Multidimensional Neural Networks Unified Theory

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Copyright © 2008, New Age International (P) Ltd., Publishers Published by New Age International (P) Ltd., Publishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailed to [email protected]

ISBN (13) : 978-81-224-2629-8

PUBLISHING FOR ONE WORLD

NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.newagepublishers.com

Dedicated to the memory of JESUS CHRIST and SARASWATI

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Preface This book deals with a novel paradigm of neural networks, called multidimensional neural networks. It also provides comprehensive description of a certain unified theory of control, communication and computation. This book can serve as a textbook for an advanced course on neural networks or computational intelligence/cybernetics. Both senior undergraduate and graduate students can get benefit from such a course. It can also serve as a reference book for practicising engineers utilizing neural networks. Further more, the book can be used as a research monograph by neural network researchers. In the field of electrical engineering, researchers have innovated sub-fields such as control theory, communication theory and computation theory. Concepts such as logic gates, error correcting codes and optimal control vectors arise in the computation, communication and control theories respectively. In one dimensional systems, the concept of error correcting codes, logic gates are related to neural networks. The author, in his research efforts showed that the optimal control vectors (associated with a one dimensional linear system) constitute the stable states of a neural network. Thus unified theory is discovered and formalized in one dimensional systems. Questioning the possibility of logic gates operating on higher dimensional arrays resulted in the discovery as well as formalisation of the research area of multi/infinite dimensional logic theory. The author has generalised the known relationship between one dimensional logic theory and one dimensional neural networks to multiple dimensions. He has also generalised the relationship between one dimensional neural networks and error correcting codes to multidimensions (using generator tensor). On the way to unification in multidimensional systems the author has discovered and formalised the concept of tensor state space representation of certain multidimensional linear systems. It is well accepted that the area of complex valued neural networks is a very promising research area. The author has proposed a novel activation function called the complex signum function. This function has enabled proposing a complex valued neural associative memory on the complex hypercube. He also proposed novel models of neuron (such as linear filter model of synapse). This book contains 10 chapters. The first chapter provides an introduction to the unified theory of control, communication and computation. Chapter 2 introduces a mathematical

(viii)

Preface

model of multidimensional neural networks and the associated convergence theorem. In Chapter 3, the concepts of multidimensional error correcting codes, multidimensional neural networks and optimization of multi-variate polynomials (associated with a tensor) over various subsets of multidimensional lattice, are related from different view points. In Chapter 4, Tensor State Space Representation (TSSR) of certain multidimensional linear systems is discussed. In Chapter 5, Unified Theory of Control, Communication and Computation in multidimensional linear systems is summarized. In chapter 6, the author proposes a novel complex signum function. In Chapter 7, a novel optimal filtering problem associated with a one dimensional linear system is formulated and solved. In Chapter 8, a linear filter model of synapse is proposed. Also a novel continuous time associative memory and the associated convergence theorem are discussed. In Chapter 9, a novel model of neuron and associated real/complex neural networks are proposed. Finally in Chapter 10, advanced theory of evolution based on the unified theory is briefly discussed. The Chapters in this book are organised in such a way that there is considerable flexibility in its use by its readers. For instance, Chapters 1 to 5 can form the basis for a graduate course on multidimensional neural networks and unified theory. This course is a compulsory course for students interested in doing research on computational intelligence (cybernetics). The students/researchers interested in doing research on complex valued neural networks will find interesting material in Chapters 6 and 9. Further, the students/ researchers interested in exploring interrelationship between signal processing and neural networks will enjoy understanding the material in Chapters 7 and 8. Finally, Chapter 10 will provide counter-intuitive insights into the theory of organic evolution. This writing project would not be possible without the cooperation of my brother Dr. G.V.S.R. Prasad and my beloved mother. I thank many colleagues at IIIT and those around the world who believe that this book is my first masterpiece. I specially thank Sri Damodaran and other employees of New Age International (P) Ltd. for making my dream of publishing this book a reality. G. Rama Murthy

Contents PREFACE 1.

INTRODUCTION

L OGICAL BASIS FOR COMPUTATION L OGICAL BASIS FOR CONTROL L OGICAL BASIS OF COMMUNICATION A DVANCED THEORY OF EVOLUTION 2.

MULTI/INFINITE DIMENSIONAL NEURAL NETWORKS, MULTI/INFINITE DIMENSIONAL LOGIC THEORY

2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 3.

INTRODUCTION MATHEMATICAL MODEL OF MULTIDIMENSIONAL NEURAL NETWORKS CONVERGENCE THEOREM FOR MULTIDIMENSIONAL NEURAL NETWORKS MULTIDIMENSIONAL LOGIC THEORY, LOGIC SYNTHESIS INFINITE DIMENSIONAL LOGIC THEORY: INFINITE DIMENSIONAL LOGIC SYNTHESIS N EURAL NETWORKS, LOGIC THEORIES, CONSTRAINED STATIC OPTIMIZATION CONCLUSIONS

MULTI/INFINITE DIMENSIONAL CODING THEORY: MULTI/INFINITE DIMENSIONAL NEURAL NETWORKS—CONSTRAINED STATIC OPTIMIZATION

3.1. 3.2. 3.3. 3.4.

INTRODUCTION MULTIDIMENSIONAL NEURAL NETWORKS: MINIMUM CUT COMPUTATION IN THE CONNECTION STRUCTURE: GRAPHOID CODES MULTIDIMENSIONAL ERROR CORRECTING CODES: ASSOCIATED ENERGY FUNCTIONS—GENERALIZED NEURAL NETWORKS MULTIDIMENSIONAL ERROR CORRECTING CODES: RELATIONSHIP TO STABLE STATES OF ENERGY FUNCTIONS

3.5. 3.6. 3.7. 3.8.

N ON-BINARY LINEAR CODES N ON-LINEAR CODES CONSTRAINED STATIC OPTIMIZATION CONCLUSIONS

(vii) 1

3 3 4 6

9

9 11 14 17 20 23 25

27

27 29 34 39 42 45 53 59

(x) 4.

Contents TENSOR STATE SPACE REPRESENTATION: MULTIDIMENSIONAL SYSTEMS

61

4.1. 4.2.

61

4.3. 4.4.

INTRODUCTION STATE OF THE ART IN MULTI/ INFINITE DIMENSIONAL STATIC/ DYNAMIC SYSTEM THEORY: REPRESENTATION BY TENSOR LINEAR OPERATOR STATE SPACE REPRESENTATION OF CERTAIN MULTI/ INFINITE DIMENSIONAL DYNAMICAL SYSTEMS: TENSOR LINEAR OPERATOR MULTI/ INFINITE DIMENSIONAL SYSTEM THEORY: LINEAR DYNAMICAL SYSTEMS – STATE SPACE REPRESENTATION BY TENSOR LINEAR OPERATORS

4.5. 4.6. 4.7. 5.

STOCHASTIC DYNAMICAL SYSTEMS DISTRIBUTED DYNAMICAL SYSTEMS CONCLUSIONS

UNIFIED THEORY OF CONTROL, COMMUNICATION AND COMPUTATION: MULTIDIMENSIONAL NEURAL NETWORKS

5.1. 5.2. 5.3. 5.4.

INTRODUCTION ONE DIMENSIONAL LOGIC FUNCTIONS, CODEWORD VECTORS, OPTIMAL CONTROL VECTORS: ONE DIMENSIONAL NEURAL NETWORKS OPTIMAL CONTROL TENSORS: MULTIDIMENSIONAL NEURAL NETWORKS MULTIDIMENSIONAL SYSTEMS: OPTIMAL CONTROL TENSORS, CODEWORD TENSORS AND SWITCHING FUNCTION TENSORS

5.5 6.

COMPLEX VALUED NEURAL ASSOCIATIVE MEMORY ON THE COMPLEX HYPERCUBE

6.1. 6.2. 6.3 6.4. 7.

8.

CONCLUSIONS

INTRODUCTION FEATURES OF THE PROPOSED MODEL CONVERGENCE THEOREMS CONCLUSIONS

63 65 69 70 73 76

79

79 80 82 90 92 95

95 96 97 105

OPTIMAL BINARY FILTERS: NEURAL NETWORKS

107

7.1. 7.2. 7.3. 7.4.

107 107 113 114

INTRODUCTION OPTIMAL SIGNAL DESIGN PROBLEM: SOLUTION OPTIMAL FILTER DESIGN PROBLEM: SOLUTION (DUAL OF SIGNAL DESIGN PROBLEM) CONCLUSIONS

LINEAR FILTER MODEL OF A SYNAPSE: ASSOCIATED NOVEL REAL/COMPLEX VALUED NEURAL NETWORKS

117

8.1. 8.2. 8.3 8.4.

117 118 120 121

INTRODUCTION C ONTINUOUS TIME PERCEPTRON AND GENERALIZATIONS A BSTRACT MATHEMATICAL STRUCTURE OF NEURONAL MODELS FINITE IMPULSE RESPONSE MODEL OF SYNAPSES: NEURAL NETWORKS

Contents

8.5. 8.6. 8.7. 8.8. 9.

10.

(xi)

N OVEL CONTINUOUS TIME ASSOCIATIVE MEMORY MULTIDIMENSIONAL GENERALIZATIONS G ENERALIZATION TO COMPLEX VALUED NEURAL NETWORKS (CVNNS) CONCLUSIONS

122 125 125 126

NOVEL COMPLEX VALUED NEURAL NETWORKS

129

9.1. 9.2 9.3. 9.4. 9.5. 9.6. 9.7. 9.8.

129 130 133 133 134 135 135 136

INTRODUCTION DISCRETE FOURIER TRANSFORM: SOME COMPLEX VALUED NEURAL NETWORKS COMPLEX VALUED PERCEPTRON N OVEL MODEL OF A NEURON: ASSOCIATED NEURAL NETWORKS CONTINUOUS TIME PERCEPTRON LEARNING LAW SOME IMPORTANT GENERALIZATIONS SOME OPEN QUESTIONS CONCLUSIONS

ADVANCED THEORY OF EVOLUTION OF LIVING SYSTEMS

137

10.1. 10.2. 10.3. 10.4.

137 137 138 139

UNIFIED THEORY: CYBERNETICS ORGANIC EVOLUTION EVOLUTION OF LIVING SYSTEMS: INNOVATIVE PRINCIPLES CONCLUSIONS

INDEX

141

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CHAPTER

1

Introduction

Ever since the dawn of civilization, the homo-sapien animal unlike other lower level animals was constantly creating tools that enabled the community to not only take advantage of the physical universe but also develop a better understanding of the physical reality through the discovery of underlying physical laws. The homo-sapien, like other lower level animals had two primary necessities: metabolism and reproduction. But, more important was the obsession with other developed necessities such as art, painting, music and sculpture. These necessities naturally lead to the habit of concentration. This most important habit enabled him to develop abstract tools utilized to study nature in most advanced civilizations. Thus the homo-sapien animal achieved the distinction of being a higher animal compared to the other animals in nature. In ancient Greece, the homo-sapien civilization was highly advanced in many matters compared to all other civilizations. Such a lead was symbolized by the development of mathematics subject in various important stages. The most significant indication of such development is left to posterity in the form of 13 books called, Euclid’s Elements. These books provide the first documented effort of axiomatic development of a mathematical structure such as the Euclidean geometry. Also, Greek, Babylonian civilizations made important strides in algebra: solving linear, quadratic equations and studying the quadratic homogeneous forms in two variables (for conic sections). Algebra was revived during the Renaissance in Italy. In algebra, solution of cubic, quartic equations was carried out by the Italian algebraists. This constituted the intellectual heritage, cultural heritage along with religious, social traditions. To satisfy the curiosity of observing the heavens, various star constellations, astronomical objects were classified. In navigating the ships for battle purposes as well as trade, astronomical observations were made. These provided the first curious data related to the natural world. In an effort to understand the non-living material universe, homosapiens have devised various tools: measuring equipment, experimental equipment, mathematical procedures, mathematical tools etc.

2

Multidimensional Neural Networks: Unified Theory

With the discovery that Sun is the center of our relative motion system by Copernicus, Ptolemaic theory was permanently forsaken. It gave Galileo, the curious motivation for deriving the empirical laws of far flung significance in natural philosophy/natural science/physics. Kepler after strenuous efforts derived the laws of planetary motion leading to some of the laws of Newton. Issac Newton formalized the laws of Galileo by developing calculus. He also developed a theory of gravitation based on the empirical laws of Kepler. Michael Faraday derived the empirical laws of electric and magnetic phenomena. Though Newton’s mechanical laws were successfully utilized to explain heat phenomenon, kinetic theory of gases as being due to mechanical motion of molecules, atoms, they were inadequate for electrical phenomena. Maxwell formalized Faraday’s laws of electro-magnetic induction leading to his field equations. Later physics developed at a feverish pace. These results in physics were paralleled by developments in other related areas such as chemistry, biology etc. Thus, the early efforts of homo-sapiens matured into a clearer view of the non-living world. The above description summarizes the pre 20th century development of this progress on homo-sapien contributions to understanding the nonliving material universe. In making conclusive statements on the origin and evolution of physical reality, the developments of the 20th century are more important. In that endeavor, Einstein’s general theory of relativity was one of the most important cornerstones of 20th century physics. It enabled him to develop a general, more correct theory of gravitation, outdating the Newtonian theory. It showed that gravitation is due to curvature of spacetime continuum. The general theory of relativity also showed that all natural physical laws are invariant under non-linear transformations. This result was a significant improvement over special theory of relativity, where he showed that all natural physical laws are invariant under linear Lorentz transformations. This result (in special theory of relativity) was achieved when Einstein realized that due to finiteness of velocity of light, one must discard the notions of absolute space and time. They must be replaced by the notions of space-time continuum i.e. space and time are not independent of one another, but are dependent. Thus, special and general theories of relativity constrained the form of natural physical law. In the 20th century, along with the Theory of Relativity, Quantum Mechanics was developed due to the efforts of M.Planck, E. Schrodinger and W. Heisenberg. This theory showed that the electromagnetic field at the quantum level was quantized. This, along with, wave-particle duality of light was considered irreconcilable with the general theory of relativity. To reconcile general theory of relativity with various quantum theories, Y. Nambu proposed a string model for fundamental particles and formalized the dynamics of light string. Utilizing the experimentally verified quantum theories of chromodynamics, electrodynamics, supersymmetry of fundamental particles (unifying Bosons and Fermions), it was possible to supersymmetrize the string model of fundamental particles, resulting in the so-called superstring (supersymmetric string)

Introduction

3

theory. Currently, to explain the non-living universe, string model hopes to be experimentally verifiable, theoretically viable model. But the material universe consists of living universe as well as non-living universe. All efforts in science probed the non-living universe using experimental as well as theoretical methodology. The efforts of all scientists enabled them to see farther by “standing on the shoulders of earlier giants”. The homo-sapien animals by devising various tools discovered and formalized various laws and theories related to non-living physical reality based dynamical systems. The homo-sapien animal learned to build machines to facilitate his life and that of the community surrounding him. By understanding the mechanism of various functional units in living system such as ear, eye, various machines such as telephone, television, loud-speaker were built. Also, in the research area of artificial intelligence in Electrical Engineering, various functions of human brain are simulated in machines called robots. In the case of living universe, the scene was entirely different. The author made various pioneering innovations on living systems unlike the extended, stretched over effort of non-living systems by various eminent scientists. The objective/goal of this is to provide artificial/manufacturable models of living systems i.e. robots which resemble in every respect living systems. In arriving at artificial models, the effort of various eminent mathematicians, scientists culminating in those of N. Wiener (who coined the word CYBERNETICS) were helpful. The important discovery and the associated formalization belonged to the pioneering efforts of the author.

LOGICAL BASIS FOR COMPUTATION George Boole developed the algebra when the variables assume “true” or “false” values. This algebra is called the Boolean algebra. Certain elementary Boolean algebraic expressions are realized in equipment called “logic gates”. When the logic gates are combined/coordinated, arbitrary Boolean algebraic expression can be computed. The combination of Boolean logic gates ( an assemblage with some minimum configuration of gates) and memory elements forms an arithmetic unit. When such a unit is coupled with a control unit the Central Processor Unit (CPU) in a computer is realized. The CPU in association with a memory, input and output units forms a computational unit without intelligence. This is just a machine which can be utilized to perform computational tasks in a fast manner. Various thought provoking modifications make it operate on data in an efficient manner and provide computational results related to various problems.

LOGICAL BASIS FOR CONTROL Faraday conducted the experiments related to electrical and magnetic phenomenon. He discovered the laws of electro-magnetic induction. Based on his investigations, Fleming

4

Multidimensional Neural Networks: Unified Theory

discovered that a time varying electric field leads to magnetic field which can be capitalized for the motion of a neutral body. He also discovered that a time varying magnetic field leads to electric field inside a neutral conductor and flow of current takes place. These formed the Fleming’s left hand and right hand rules relating the relativistic effects between the electric field, magnetic field and conductor. These investigations of Faraday and other scientists naturally paved the way for electric circuits consisting of resistors, inductors and capacitors. Such initial efforts led to canonical circuits such as RL circuit, RLC circuit, RC circuit etc. The systems of differential equations and their responses were computed utilizing the analytical techniques. The ability to control the motion of an arbitrary neutral object led to applications of electrical circuits and their modifications for control of trajectories of aircrafts. Thus, the automata which can perform CONTROL tasks was generated. These control automata were primarily based on electrical circuits and operate in continuous time with the ability to make synchronization at discrete instants. Later utilizing the Sampling Theorem, sample-data control systems operating in discrete time were developed.

LOGICAL BASIS OF COMMUNICATION The problem of communication is to convey message from one point in space to another point in space as reliably as possible. The message on being transmitted through the channel, by being subject to various forms of disturbance (noise) is changed/garbled. By coding the message (through addition of redundancy), it is possible to retrieve the original message from the received message. Thus, the three problems: control, communication and computation can be described through the illustration in Figure 1.1. From the illustration, the message that is generated may be in continuous time or discrete time. Utilizing the Sampling Theorem, if the original signal is band-limited, then the message can be sampled. The sampled signal forms the message in discrete time. The message is then encoded through an encoder. It is then transmitted through a channel. If the channel is a waveform channel, various digital modulation schemes are utilized in encoding. The signal, on reaching the receiver is demodulated through the demodulator and then it is decoded. This whole assembly of hardware equipment forms the COMMUNICATION equipment. The above summary provided the efforts of engineers, scientists and mathematicians to synthesize the automata which serve the purpose of CONTROL, COMMUNICATION AND COMPUTATION. These functions are the basis of automata that stimulate living systems. These automata model the living systems. In other words, control, communication and computation automata when properly assembled and co-ordinated lead to robots which simulate some functions of various living systems. In the above effort at simulating the functions of living systems in machines, traditionally the control, communication and computation automata led to sophisticated robots (which served the purpose pretty well). Thus, the utilitarian viewpoint was partially satisfied. But, the author took a more FUNDAMENTAL approach to the problem of simulating a

Multidimensional Neural Networks: Unified Theory

6

are extended to multi/infinite dimensional linear systems. Also, the results developed in one dimension for computation of optimal control are immediately extended to certain multi/infinite dimensional linear systems. This result in association with the formalization of multi/infinite dimensional logic theory, multi/infinite dimensional coding theory (as an extension of one dimensional linear and non-linear codes) provided the formal UNIFIED THEORY in multi/infinite dimensional linear systems. The formal mathematical detail on models of living system functions are provided in Chapters 2 to 5. These chapters provide the details on control, communication and computation automata in multiple dimensions. Several generalized models of neural networks are discussed in Chapters 5 to 9. Also relationship between neural networks and optimal filters is discussed in Chapter 7. In Chapter 10 advanced theory of evolution is discussed.

ADVANCED THEORY OF EVOLUTION Mathematical models of living system functions motivated us to take a closer look at the functions of natural living systems observed in physical reality. In physical reality, we observe homo-sapiens as well as lower level animals such as tigers, lions, snakes etc. It is reasoned that some of the functions of natural living systems are misunderstood or un-understood. Biological living systems such as homo-sapiens lead to a biological culture. In a biological culture that originated during the ice age in oceans, various living species were living in the oceans. Through some process, the two necessities of metabolism and reproduction were developed by all living species. The homo-sapien species was responsible for our current understanding of various activities, functions of observed living systems. The author hypothesizes that the homo-sapien interpretations are totally wrong. For instance, • Metabolism which leads to killing of one species by another is unnecessary to sustain life. • The belief (like many superstitions) that death and aging are inevitable is only partially true. To be more precise, it should be possible to take non-decayed organs of a living species and by recharging the dead cells, make it living. Many such innovative ideas on living systems are discussed in Chapter 10. The only necessities of natural living systems that are observed are ‘metabolism’ and ‘reproduction’. By and large the only organization and community formation that we see in other (than homo-sapiens) natural sustems are of the following form • • • • •

Migratory pattern of birds Sharing the information on the place of food Forming a group of families to satisfy the reproductive needs Occasional bird songs of mutual courtship Occsional rituals related to protecting the members of their group etc.

Introduction

7

The organization, culture observed in other biological systems and other natural living systems is nowhere comparable to those observed in the homo-sapien species. But the author hypothesizes that this marginal/poor organization is primarily due to lack of coordination which is achieved through the language. Thus, major effort in organizing the lower level species of living systems is through teaching a language. Thus, organization of living systems other than the homo-sapiens (for homo-sapien and other purposes) should be possible. An important part of organizing the homo-sapiens was the educational system through an associated language. In the same spirit, by teaching some lower level animals to speak certain language, they could be organized/educated to understand as well as develop science and technology. When the lower level animals are organized in a zoo through various methods, they could lead to a culture and a civilization. Various natural living machines have developed organs/functional units due to evolutionary needs. These functional units essentially include sensors to collect video, audio information or more generally sensors to collect data on the surrounding environment in the universe. The data gathered by the living machine from the surrounding environment in physical reality is utilized to perform some primary functions such as metabolism, reproduction etc. The data is processed by various functional sub-units inside the brain of a living machine. Thus the understanding of the operation of various functional sub-units in the brain of natural living machines leads to building artificial living machines which are far superior in functional capabilities.

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CHAPTER

2

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

2.1 INTRODUCTION One dimensional logic theory is concerned with the study of static/dynamic transformations on one dimensional arrays of zeroes and ones to arrive at arrays of zeroes and ones. Various standard logic gates such as AND, OR, NOT, NAND, XOR, NOR are defined on one dimensional arrays/vectors. The logic synthesis of digital integrated circuits, consisting of the interconnection of logic gates which transit through a set of states, is performed through the utilization of the associated state transition diagram. The set of allowed transitions in the state space lead to various classes of digital circuits such as shift registers, counters, flip flops etc. In one dimensional logic theory various theorems on the decomposition, synthesis of Boolean functions are proved and are utilized in the logic synthesis of complex digital integrated circuits. In the practical implementation of such digital integrated circuits, semiconductor technology with devices such as diodes, transistors, field effect transistors was effectively utilized. The design and implementation of complex digital integrated circuits led to the development of highly sophisticated computers, computer systems serving various practical applications. Some practical applications such as those in medical imaging, remote sensing, pattern recognition led to the design and implementation of various types of parallel computers. These computers operate on two dimensional arrays of zeroes and ones. But the processing units in these computers treat the two dimensional array elements as those from one dimensional arrays. Thus, the two dimensional nature of an array with dependency structure is never capitalized. This limitation led the author to innovate information processing units which operate on two/multidimensional arrays. Such information processing units should necessarily be based on sub-units which operate on arrays of binary data and produce binary arrays. These sub-units constitute the two/multidimensional logic circuits. A more

10

Multidimensional Neural Networks: Unified Theory

general class of information processing sub-units and thus the units operate on arrays whose entries are allowed to assume multiple (not necessarily binary) values. Automata which operate on multidimensional arrays to perform desired operation can be defined heuristically in many ways. In some applications such as in 3-d array/ image processing, the information processing operation can only be defined heuristically based on the required function. But, a more organized approach to define multidimensional logic functions is discovered and formalized by the author. In this chapter, the author describes the mathematical formalization for multidimensional logic units. The relationship between multidimensional logic units and multidimensional neural networks is also discussed. The generalization of the results to infinite dimensions is also briefly described. Two dimensional neural networks were utilized by various researchers working in the area of neural networks. The application of two dimensional neural networks to various real world problems was also extensively studied. But, an effective mathematical abstraction for modeling two/multi/infinite dimensional neural networks was lacking. The author in this chapter demonstrates that tensors provide a mathematical abstraction to model multi/ infinite dimensional neural networks. The contents of this chapter are summarized as follows: A mathematical model of an arbitrary multidimensional neural network is developed. A convergence theorem for an arbitrary multidimensional neural network represented by a fully symmetric tensor is stated and proved. The input and output signal states of a multidimensional logic gate/neural network are related through an energy function, defined over the fully symmetric tensor representing the multidimensional logic gate, such that the minimum/maximum energy states correspond to the output states of the logic gate realizing a logic function. Similarly, a logic circuit consisting of the interconnection of logic gates, represented by a symmetric tensor, is associated with a quadratic/higher degree energy function. Multidimensional logic synthesis is described. Infinite dimensional logic theory, logic synthesis are briefly discussed through the utilization of infinite dimension/ order tensors. This chapter is organized as follows. In section 2, a mathematical model of an arbitrary multidimensional neural network and associated terminology is developed. In section 3, a convergence theorem for an arbitrary multidimensional neural network is proved. In section 4, the input/stable states of a multidimensional neural network are associated with the input/output signal states of a multidimensional logic gate. A mathematical model of an arbitrary multidimensional logic gate/circuit is described. Thus, multidimensional logic theory, logic synthesis is formalized. In section 5, infinite dimensional logic theory, logic synthesis are described. In section 6, the relationship between multidimensional neural networks, multidimensional logic theories, various constrained static optimization problems is elaborated. Various constrained optimization problems that commonly arise in various problems are listed. Various innovative ideas in multidimensional neural networks are briefly described. The chapter concludes with a set of conclusions.

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

11

2.2 MATHEMATICAL MODEL OF MULTIDIMENSIONAL NEURAL NETWORKS A discrete time multidimensional neural network paradigm is a dynamical system evolving in discrete time. It can be represented by a weighted connectionist structure in multidimensions. Thus, there is a weight attached to each edge of the connectionist structure in multidimensions and a threshold value attached to each node. At each node of the connectionist structure, a certain algebraic threshold function is computed. It is well known in the theory of one dimensional neural networks that a symmetric matrix can be utilized to represent a one dimensional neural network. With the motivation, applications of one dimensional neural networks, two dimensional neural networks were heuristically designed and utilized for various applications. But, the author for the first time realized that tensor is the most natural mathematical abstraction that can be utilized to represent two/multidimensional neural networks.

Multidimensional Neural Networks: Tensors Before describing the mathematical model of multidimensional neural networks, the following discussion on tensors and associated concepts is very relevant. It is important to realize that given n independent variables, the expression n

∑C X i =1

i

(2.1)

1

is called a homogeneous linear form in the variables, the expression n

n

∑∑ C i =1 j =1

ij

Xi X j

(2.2)

is called a homogeneous quadratic form, the expression n

n

n

∑∑∑ C i =1 j =1 k =1

ijk

Xi X j K k

(2.3)

is called a homogeneous form (BoT) of degree three and so on. Given the components of a tensor of order n, of dimension m , it is possible to define a homogeneous form of degree n. The connection structure of a one dimensional neural network, the symmetric matrix, is naturally associated with a homogeneous quadratic form as the energy function, which is optimized over the one dimensional hypercube. Thus, in one dimension, to utilize a homogeneous form of degree n as the energy function, a generalized neural network is employed, in which, at each neuron, an arbitrary algebraic threshold function is computed. But, in multidimensions, to describe the connection structure of a neural network, a tensor is necessarily utilized.

Multidimensional Neural Networks: Unified Theory

12

With the above description of necessity of tensors to represent generalized/ multidimensional neural networks, some notation related to tensors is provided to facilitate the description of mathematical model of an arbitrary multidimensional neural network.

Tensors, Tensor Products Matrices are utilized to represent quadratic forms, whereas tensors are necessary to represent a homogeneous form of degree n. i.e.

Suppose, one second order tensor is a linear function of another second order tensor Aik = λiklm Bim

(2.4)

where λiklm is a set of k 4 coefficients. It is easy to see that λiklm is a tensor of dimension k and order 4. This is illustrative of linear transformation of tensors. Now, we discuss some concepts in the multiplication of tensors. Let A i k and Bi k be the components of two second order tensors. Consider all possible products of the form Ciklm = Aik Bim

(2.5)

Then, the numbers C iklm are the components of a fourth-order tensor, called the outer product of tensors with components A ik and Bi k. Multiplication of any number of tensors of arbitrary order is defined similarly (BoT), i.e. the product of two or more tensors is the product of the components of the tensors, which are factors. The order of a tensor product is clearly the sum of the orders of the factors. Contraction of Tensors: The operation of summing a tensor of order n (n >2) over two of its indices is called contraction. It is clear that contraction of a tensor of order n leads to a tensor of order n-2. This tensor can be repeatedly contracted to arrive at a tensor of order 2 or a scalar depending on whether n is even or odd. The result of multiplying two or more tensors and then contracting the product with respect to indices belonging to different factors is often called an Inner Product of the given tensors. Thus, based on the notation associated with the indices, it is understood from the context whether inner product or outer product of tensors is utilized. With the above requisite notation from tensor algebra summarized, before describing a mathematical model of an arbitrary multidimensional neural network, the following intuitive discussion is provided to facilitate easier understanding. The state of a neuron at the discrete time instant n+1 is computed by summing the contributions from other neurons connected to it through synaptic weights which are the components of a fully symmetric tensor S, representing the connection structure and the

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

13

state tensor of neuronal states at the time instant n. Thus, we first compute the outer product of connection tensor and the state tensor of neurons at the time instant n and perform the contraction over all the indices (representing the neurons) connected to a chosen neuron. Thus, this inner product operation followed by determining its sign/parity/polarity (positive or negative value) gives us the state tensor at time instant n+1. This procedure is repeated at all the neurons where the state is updated. Remark Throughout the research article, the notation “multidimensional neural network” is utilized. The standard notation associated with tensors utilizes the term, “dimension” to represent the number of values an independent variable can assume and the term, “order” to represent the number of independent variables. Thus, the state tensor order represents the number of independent dimensions in the multidimensional neural network, MN. The notational confusion between the usage of terms “order”, “dimension” should be resolved from the context.

Mathematical Model Description Let MN be a multidimensional neural network of dimension m and order n, then MN is uniquely specified by ( S, T ) where ( the number of neurons in each independent variable/ dimension/ order index is m ) S is a fully symmetric tensor of order 2n and dimension m . S, the connection structure of multidimensional neural network, is a fully symmetric tensor in the following sense

Si 1, i 2,..., in ; j 1, j 2,..., jn = Sj 1, j 2,..., jn ; i 1, i 2,...,in

(2.6)

for all {i1,i2,...,in}, {j1,j2,...,jn}. This captures the intuitive notion that the multidimensional neural network has nodes which correspond to the multidimensional neurons. The connectionist structure of the network, in the fully connected case, has a synaptic connection from every neuron to every other neuron and thus specifies the number of order indices/ dimensions/variables of the fully symmetric tensor. Furthermore, it is fully symmetric since there is a link between any two nodes and the weight attached to the link is the same in both directions. T is a tensor compatible with S such that each component is the threshold at the node (i1, i2,...,in) of the multidimensional neural network. Every node ( multidimensional neuron ) can be in one of the two possible states, either +1 or –1. The state of node (i1, i2,...., in) at time t is denoted by Xi1, i2,..., in (t). The state of MN at time t is the tensor Xi1, i2,..., in (t), where X is tensor of dimension m and order n. The state evolution at node (i1, i2,...,in) is computed by

Xi 1, i 2,..., in (t + 1) = Sign ( Hi 1, i 2,..., in (t)),

(2.7)

Multidimensional Neural Networks: Unified Theory

14

where, m

m

j1= 1

jn = 1

Hi 1, i 2,..., in (t) = ∑ ... ∑ Si 1,..., in ; j 1,..., jn X j 1,..., jn (t) − Ti 1,..., in (t)

(2.8)

The next state of the network Xi1,...,in (t +1) is computed from the current state by performing the evaluation (2.7) at a subset of the nodes of the multidimensional neural network, to be denoted by G. The modes of operation of the network are determined by the method by which the subset G is selected in each time interval. If the computation is performed at a single node in any time interval, i.e.|G| = 1, then we will say that the network is operating in a serial mode, and if |G| = m n, then we will say that the network is operating in a fully parallel mode. All other cases, i.e. 1 < |G| < m n, will be called parallel modes of operation. Unlike a one dimensional neural network, multidimensional neural network lends itself for various parallel modes of operation. It is possible to choose G to be the set of neurons placed in each independent dimension or a union of such sets. The set G can be chosen at random or according to some deterministic rule. A state of the network is called stable if and only if

Xi 1,..., in (t) = Sign (S ⊗ Xi 1,..., in (t) − Ti 1,...iin )

(2.9)

where ⊗ denotes inner product i.e. outer product followed by contraction over the appropriate indices. Once the network reaches such a state, there is no further change in the state of the network no matter what the mode of operation is.

2.3 CONVERGENCE THEOREM FOR MULTIDIMENSIONAL NEURAL NETWORKS Utilizing a fully symmetric tensor to represent the connection structure of a multidimensional neural network, utilizing the notation of tensor products, in the following, convergence theorem for an arbitrary multidimensional neural network is stated and proved. Theorem 2.1: Let MN = (S, T) be a multidimensional neural network of dimension m and order n. S is a fully symmetric tensor of order 2n and dimension m with Si 1,..., in ; i 1,...,in ≥ 0 . The network MN always converges to a stable state while operating in the serial mode (i.e. there are no cycles in the state space) and to a cycle of length utmost 2 while operating in a fully parallel mode (i.e. the cycles in the state space are of length ≤ 2 ). Proof: Serial mode of operation of the multidimensional neural network is first considered. In this mode of operation, during each time step of the operation of the neural network, the state of only one neuron is updated. In other words, the state of each neuron is only updated serially. At each multidimensional neuron in the network MN, the total synaptic

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

15

contribution from all neurons is first determined and its sign is determined to arrive at the updated state of the neuron. Mathematically, this is achieved by computing the outer product of the fully symmetric tensor S and the {+1, –1} state tensor of the multidimensional neural network. In tensor notation, this is specified by

Ci 1,..., in ; j 1,..., jn = Si 1,..., in ; j 1,..., jn X j 1,..., jn .

(2.10)

The total synaptic contribution at any neuron located at the location (i1, i2,..., in) is determined by contracting the above outer product over all the indices {j1, j2,..., jn} i.e. over all the neurons connected to it through the synaptic weights determined by the components of the fully symmetric tensor S. The resultant scalar synaptic contribution at any neuron (i1, i2,..., in) is thus determined by the inner product operation. The sign of the resulting scalar constitutes the updated state of neuron. Thus, the state of any neuron (i1, i2,..., in) in the multidimensional neural network in the serial mode of operation is given by m

m

j1= 1

jn = 1

Xi 1, i 2,..., in ( k + 1) = Sign ( ∑ ... ∑ Ci 1,..., in ; j 1,..., jn ( k ) − Ti 1,..., in )

= Sign (S ⊗ X(k ) − T )

(2.11) (2.12)

where ⊗ is utilized as the symbol to denote the inner product between compatible tensors. This symbol is sometimes suppressed and it should be understood from the context whether inner product/outer product between the tensors is meant. With the state updating scheme in the tensor notation specified, the energy function that is optimized in the network MN is described. It is given by m

m

m

m

E = < X( k ), S ⊗ ( k ) > = ∑ .. ∑ ∑ .. ∑ Si 1,..., in ; j 1,..., jn X i 1,..., in ( k ) X j 1,..., jn ( k ) i 1 = 1 in = 1 j 1 = 1

jn = 1

(2.13)

where < > denotes the inner product operator between the compatible tensors. It is assumed in the above specification of the energy function of the neural network MN that the threshold at each neuron is zero. This is no loss of generality, since by augmenting the tensor S and the state tensor, the threshold values can be forced to be zero. It is easy to see that such a thing can always be done by considering a one dimensional neural network in which the threshold at each neuron is non-zero and arriving at a network in which the threshold at each neuron can be made zero by augmenting the state vector as well as the connection matrix. Utilizing the definition of the above energy function of the network, let ∆E = E1 ( t + 1) − E1 ( t) , (discrete time index t instead of k is used) be the difference in the energy associated with two consecutive states (transited in the serial mode of operation of the multidimensional neural network ), and let ∆X i 1,....in denote the difference between the next state and the current state of the node at location (i1, i2,..., in) at some arbitrary time t. Clearly,

Multidimensional Neural Networks: Unified Theory

16

0, if, Xi1,..., in (t) = Sign (Hi1,...,in (t)) ∆Xi 1,..., in = {−2, if, Xi 1,..., in (t) =1, and, Sign( Hi 1,...,in (t)) =− 1

(2.14)

+2, if, X i 1,..., in (t ) = −1, and, Sign ( H i 1,..., in (t ) = + 1 By assumption, the computation (2.14) is performed only at a single node at any given time. Suppose this computation is performed at any arbitrary node at location (i1, i2,..., in) ; then the difference in energy resulting from updating the network state is given by

..∑ Si 1,..., in ; j 1,..., jn X j 1,..., jn + ∑ ..∑ Si 1,...,in ; j 1,..., jn Xi 1,...,in ) ∆E = ∆Xi 1,..., in (∑ j1 jn i1 in

+ Si 1,..., in ; i 1,...,in ∆Xi 1,..., in − 2∆Xi 1,..., in Ti 1,..., in

(2.15)

Utilizing the fact that S is fully symmetric and the definition H i1,..., in (t), it follows that

∆E = 2 ∆Xi 1,..., in Hi 1,...,in + Si 1,...,in ;i 1+ ,...in

∆Xi 1,...., in

(2.16)

Hence, since ∆Xi 1,..., in Hi 1,..., in ≥ 0 and Si 1,..., in i 1,..., in ≥ 0 , it follows that at every time instant, ∆E ≥ 0 . Thus, since the energy E is bounded from above by the appropriate norm of S, the

value of energy will converge. Now, it is proved in the following that convergence of energy implies convergence to a stable state. Once the energy in the network has converged, it is clear from the following facts that the network will reach a stable state after utmost m 2n time intervals. (a) if ∆X = 0 then it follows that ∆E = 0; (b) if ∆X ≠ 0 , then ∆E = 0, only if the change in Xi 1,..., in (t) is from –1 to +1, with Hi 1,..., in = 0. In the fully parallel mode of operation of the network MN, the state updating scheme for the state tensor of MN is given by

Xi 1,..., in (t + 1) = Sign (S ⊗ Xi 1,..., in (t) − Ti 1,..., in )

(2.17)

where ⊗ denotes the inner product between compatible tensors. Since, the serial mode proof shows that a stable state is always reached with the above stated updating scheme, it is immediate that by pairwise flipping of the values of any two dimension variables in the state tensor, the same energy function value is attained. This, in turn implies that in the parallel mode of operation of a multidimensional neural network, either a stable state is reached or a cycle of length utmost 2 is reached (The two state tensors lead to the same value of the energy function). This approach to the proof for the parallel mode of operation follows the one provided in reference. [Br G] Q. E. D

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17

2.4 MULTIDIMENSIONAL LOGIC THEORY, LOGIC SYNTHESIS One dimensional logic theory as well as logic synthesis deal with information processing logic gates, logic circuits which operate on one dimensional arrays of zeroes and ones (or more generally one dimensional arrays containing finitely many symbols ). The operations performed by AND, OR , NOR, NAND, XOR gates have appropriate intuitive interpretation in terms of the entries of the one dimensional arrays i.e. vectors. Any effort to generalize the one dimensional logic operations to multidimensions leads to various heuristic possibilities and requires considerable ingenuity in formalizing a definition. But, in the following, utilizing the multidimensional neural network model described above, a formal/ mathematical procedure to multidimensional logic theory is described. The input and output signal states of a multidimensional logic gate are related through an energy function. Equivalently, the multidimensional logic functions are associated with the local optimum of various energy functions defined over the set of input m-d arrays. In view of the mathematical model of a multidimensional neural network described in section 3, it is most logical to define the minimum/ maximum energy states of a multidimensional neural network (optimizing an energy function over the multidimensional hypercube ) to correspond to the multidimensional logic gate functions operating on the input arrays. Definition 2.1 A multidimensional logic function realized through a multidimensional logic gate (with inputs and outputs) is defined to be the local minimum/maximum of the energy function of an associated multidimensional neural network. Equivalently, the local optima of the energy function of a multidimensional neural network correspond to the logic functions that are realized through various logic gates. The following detailed description is provided to consolidate the above definition vital to multidimensional logic theory. The logic functions which operate on the input array are identified to be the stable states of a multidimensional neural network ( in multiple independent variables i.e. time, space etc.). These are the transformations between a set of input states of a multidimensional neural network which converge to a stable state on iteration of a multidimensional neural network. In other words, in multiple independent variables, the mapping between the input states and the stable states to which the network converges on iteration are defined to be the logic functions realized by a multidimensional logic gate. By the proof of the convergence theorem, the logic functions are invariants of a tensor on the multidimensional hypercube. The definition of multidimensional logic function is illustrated in Figure 2.1. In the case of one dimensional logic theory, it has been shown that the set of stable states of a neural network correspond to various one dimensional logic functions (CAB). With the definition of multidimensional logic function stated and clarified in many redundant ways above, multidimensional logic synthesis is described in the following.

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Multidimensional Neural Networks: Unified Theory

Multidimensional Logic Synthesis A multidimensional logic circuit consists of an arbitrary interconnection of multidimensional logic gates. Multidimensional logic synthesis, as in one dimension, involves synthesizing logic circuits for different purposes. In view of the above definition of multidimensional logic functions defined through the local optima of energy functions (realized through multidimensional neural networks), it is natural to see if it is possible to associate energy functions with multidimensional logic circuits. When such a scalar valued energy function can be associated with logic circuits, the problem of multidimensional logic synthesis, is reduced to realizing such energy functions. In the following, this important idea is developed. A multidimensional logic circuit consists of interconnection of multidimensional logic gates. But, the interconnection structure of a multidimensional logic gate is represented by a fully symmetric tensor. Since, every two gates in a logic circuit need not necessarily be connected to one another, a multidimensional logic circuit connection structure is represented by a tensor of necessary/compatible order which is not necessarily fully symmetric but it is required to be minimally symmetric. Thus, this block symmetric tensor which is fully symmetric within the blocks (representing the connection structure of a multidimensional neural network corresponding to a component logic gate) provides a representation of multidimensional logic circuit. This tensor is utilized to associate quadratic/higher degree energy functions with the multidimensional logic circuit. The set of local optima of the energy functions constitute the stable states of one or more interconnected logic gates. Thus, the set of input states (input pins) and output states (output pins) of an entire multidimensional logic circuit are related through an energy function, defined over the connection structure of a very high dimensional neural network. The set of local optima of the energy function relating the input and output pins of a multidimensional logic circuit realize various multidimensional logic functions. From the above description, it is evident that the multidimensional logic synthesis depends on how the multidimensional logic gates are connected to one another. The structure of interconnection determines the structure of symmetric tensor representing the multidimensional logic circuit. The essential result in multidimensional logic synthesis is summarized through the following theorem. Theorem 2.2: Given a multidimensional logic circuit, there exists a block symmetric tensor S, representing the inter-connection structure of multidimensional neural networks (modeling the multidimensional logic gates). The mapping between the input and output states of a multidimensional logic circuit corresponds to that between input tensors, local optima of energy function (quadratic/higher degree) represented by the block symmetric tensor. The stable states of interconnected multidimensional neural networks represent the multidimensional logic functions synthesized by the logic circuit.

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

19

The proof of the above theorem follows from the convergence theorem and is avoided for brevity. The classification of multidimensional logic circuits is based on the type of transitions allowed between the states in the multidimensional state space. The type of state transitions fall into the following form: (a) whether the next state reached depends on the past state only or not, as in one dimensional logic synthesis, (b) the type of neighbourhood of states about the current state on which the next state reached depends. The type of neighbourhoods about the current state are classified into few classes. These classes are similar to those utilized in the theory of random fields, multidimensional image processing, (c) the classification of trajectories transited by the multidimensional neural network or a local optimum computing circuit/scheme. In the above discussion, we considered quadratic forms as the energy functions (motivated by the simplest possible neural network model) optimized by the logic gates, which when connected together lead to logic circuits. This approach toward multidimensional logic theory motivates the definition of more ‘general‘ switching/logic functions as the local optimum of higher degree forms over the various subsets of multidimensional lattice (hypercube, bounded lattice etc.). Definition 2.2 A generalized logic function (representing a generalized logic gate or generalized logic circuit) is defined as a mapping between an m -dimensional input array and the local optimum of a tensor based form of degree greater than or equal to two, over various subsets of multidimensional lattice (the multidimensional hypercube, multidimensional bounded lattice). These local optimum of higher degree form (based on a tensor) are realized through the stable states of a generalized multidimensional neural network. In (Rama 3) , it is shown that the strictly generalized logic function defined above has better properties than the ordinary logic function described in Definition 4.1. The generalized logic function is related to a multidimensional encoder utilized for communication through multidimensional channels. Now, with the generalized multidimensional logic gate defined above, logic synthesis with these types of logic gates involves interconnection of them in certain topology. This ordinary and generalized approach to multidimensional logic gate definition and logic synthesis is depicted in Figures 2.1 to 2.3. Detailed documentation on logic synthesis and design of future information processing machines is being pursued.

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

21

Proof: One dimensional neural network with state vector size infinity is uniquely defined by (S, T) where S is an infinite dimensional (rows as well as columns) symmetric matrix and T is an infinite dimensional vector of thresholds at all the neurons. The state of the neural network at time t is a vector whose components are +1 and –1. The next state of a node is computed by

Xi (t + 1) = Sign(Hi (t)) = + 1, if, Hi (t) ≥ 0

(2.18)

–1, otherwise where, ∞

Hi (t ) = ∑ Sji X j (t ) − Ti

(2.19)

j =1

The entries of S are such that the infinite sum in the above expression converges. The next state of the network i.e. X ( t+1 ), is computed from the current state by performing the evaluation (2.18) at a subset of the nodes of the network, to be denoted by K. The mode of operation of the network is determined by the method by which the set K is selected at each time interval i.e. if |K| = 1, then we will say that the network is operating in a serial mode. Without loss of generality T = 0. In the following, we consider the serial mode of operation. We argue that with the above stated updating scheme at an arbitrary chosen neuron, the energy function (quadratic) increases. ∞

∞

E( k ) = ∑∑ Sij Xi ( k ) X j ( k )

(2.20)

i =1 j =1

Without loss of generality, consider the case where all the thresholds are set to zero. It is easy to see (set the last component of state vector to –1 and appropriately augmented entries of S) that for any finite L, we have L

L

∑∑ S i =1 j =1

ij

Xi ( k ) X j ( k ) ≤

L

L

∑∑ S i =1 j =1

ij

Xi ( k + 1) X j ( k + 1)

(2.21)

by the convergence theorem for one dimensional neural networks of order L, for any arbitrary L. Now let L tend to infinity. Hence ∞

∞

∑∑ Sij Xi (k) X j (k ) ≤ i =1 j =1

∞

∞

∑∑ S i =1 j =1

ij

Xi ( k + 1) X j ( k + 1)

(2.22)

Thus, in the serial mode, the network converges to a stable state. By the Convergence Theorem for one dimensional neural network (with the state vector size finite) in the parallel mode of operation, if any finite set of nodes is state updated, there is either convergence or existence of a cycle of length 2. Thus, when an infinite

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Multidimensional Neural Networks: Unified Theory

dimensional vector is state updated in the parallel mode, for every finite segment of it, either there is convergence or a cycle of length 2 (utmost two vectors for which the energy values are the same) exists. Since, the energy function associated with the infinite dimensional vector is the limit of those associated with the finite segments, it is evident that the scalar energy values converge or a cycle of length utmost two exists. Q.E.D. Now, we discuss briefly, the other infinite dimensional neural networks of dimension infinity and order finite/infinite ( modeling tensor variables). The following lemma is well known from the set theory. Lemma 2.1: Countable union of countable sets is countable. The above lemma implies that the convergence theorem proved above in association with the convergence theorem for multidimensional neural networks (its proof argument in section 3) provides us with the convergence proof for a large class of infinite dimensional neural networks (dimension and/or order of tensors utilized in modeling is infinity). Details on the convergence theorem for infinite dimensional neural networks are provided below. Tensors utilized to represent the connection structure, state of neurons of infinite dimensional neural network are such that the either the dimension or the order is finite/infinite with not both of them being finite (either the dimension or the order or both are infinite ). In one dimension, when the number of neurons is infinite and a quadratic energy function is optimized through a neural network scheme, by a straightforward extension of the results in (Rama 3), the stable states of the neural network constitute a graph-theoretic code (with the length of the codeword being infinite). The set over which optimization is carried out is the unbounded unit hypercube (countable number of entries in the infinite dimensional state vector), a subset of the lattice ( based on one independent variable ). The following theorem is concerned with the points on the lattice in multi/infinite dimensions. This theorem is the infinite dimensional extension of the result proved in section 3. Theorem 2.4: Let MN = (S, T) be an infinite dimensional neural network of order n/ infinite and dimension infinity (number of neurons in each dimension). S is a fully symmetric tensor of dimension infinity and order 2n/infinity with Si 1,..., in ; i 1,...,in ≥ 0 . The network MN always converges to a stable state while operating in a serial mode (i.e., there are no cycles in the state space), while in the parallel mode, the network will always converge to a stable state or to a cycle of length 2 (i.e., the cycles in the state space are of length ≤ 2). Proof: For a multidimensional neural network modeled by a tensor of dimension and order finite, in the serial mode of operation, the network always converges to a stable state. Since, the quadratic energy function is a scalar value defined over the connection tensor (whose order, dimension are finite ), by letting the dimension and/or order tend to infinity in (2.13), it is immediate that the energy function value increases in the serial mode until a stable state is reached starting in a certain initial state. Thus, for various infinite

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

23

dimensional neural networks considered, convergence to a stable state in the serial mode of operation is ensured ( i.e. there are no cycles in the state space ). In the parallel mode of operation of the infinite dimensional neural network, by the same reasoning as in Theorem (2.1), the network will always converge to a stable state or to a cycle of length 2 depending on the order of the network ( i.e. the cycles in the state space are of length less than or equal to 2). Q.E.D As in the case of multidimensional logic theory, the above convergence theorem is utilized as the basis to describe infinite dimensional logic theory as well as logic synthesis. It should be noted that the infinite dimensional logic synthesis only has theoretical importance. Brief discussion on infinite dimensional versions is provided for the sake of completeness. Definition 2.3 An infinite dimensional logic function realized through an infinitedimensional logic gate (with inputs and outputs) is defined to be the local minimum/maximum of the energy function of an associated infinitedimensional neural network. Equivalently, the local optima of the energy function of an Infinitedimensional neural network correspond to the logic functions that are realized through various logic gates. With the above definition of infinite dimensional logic function, detailed results in infinite dimensional logic synthesis are being developed along the lines of those in multidimensional logic synthesis. Brief description is provided in the following for the sake of completeness. An infinitedimensional logic circuit consists of an arbitrary interconnection of infinitedimensional logic gates. Infinitedimensional logic synthesis, as in one dimension involves synthesizing logic circuits for different purposes. These infinite dimensional logic circuits only have theoretical implementations. Infinitedimensional logic synthesis depends on how the infinitedimensional logic gates are connected to one another. The structure of interconnection determines the structure of symmetric tensor (order and/or dimension is infinity) representing the infinitedimensional logic circuit.

2.6 NEURAL NETWORKS, LOGIC THEORIES, CONSTRAINED STATIC OPTIMIZATION Multidimensional neural networks provide a computational paradigm to determine the local optima of quadratic as well as higher degree forms defined in terms of tensors (including matrices) over various subsets of the multidimensional lattice. These units which map a multidimensional array/tensor to a local optimum (stable state of the multidimensional neural network), thus constitute the multidimensional logic gates. Interconnection of such multidimensional logic gates constitutes a multidimensional logic circuit. Thus, multidimensional logic circuits are interconnected multidimensional neural

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Multidimensional Neural Networks: Unified Theory

networks. The interconnection structure weights are represented through a symmetric tensor. Thus, multidimensional logic theory/logic synthesis are associated with the theory of multidimensional neural networks. These theories are in turn related to static optimization of various forms (quadratic as well as higher degree) over different subsets of lattice and other sets. Various constrained static optimization problems that are of interest in different applications (neural networks, logic theories etc.) are summarized below: (1) Optimization of a quadratic form in finitely many variables over the one dimensional hypercube (one independent variable), (2) Optimization of a higher degree form in finitely many variables over the one dimensional hypercube (one independent variable), (3) Optimization of a quadratic form over the infinite dimensional (size of the state vector) hypercube in one dimension, (4) Optimization of a higher degree form over the infinite dimensional (size of the state vector) hypercube in one dimension, (5) Optimization of a quadratic form over the finite/infinite dimensional hypercube in finitely/infinitely many dimensions, (6) Optimization of a higher degree form over the finite/infinite dimensional hypercube in finitely/infinitely many dimensions, (7) Optimization of a quadratic form over a bounded lattice in finitely/infinitely many dimensions, (8) Optimization of a higher degree form over a bounded lattice in finitely/infinitely many dimensions, (9) Optimization of a quadratic form over the unbounded lattice in finitely/infinitely many dimensions, (10) Optimization of a higher degree form over the unbounded lattice in finitely/ infinitely many dimensions. When the constraint set is the lattice (unbounded lattice) in finitely/infinitely many dimensions and the number of state variables is not finite but countable, the objective function is a power series each of whose terms is a quadratic/higher degree form. It is proved in (Rama 3) that some of the constrained optimization problems arise in the design of multi/infinite dimensional codes. In (Rama 4), various optimization problems described above are utilized in dynamic optimization setting. In the following , various innovative themes in multi/infinite dimensional neural networks are briefly discussed.

Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory

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Continuous Time Neural Networks The well known model of a neural network is a discrete time system in one or multiple dimensions. A signal design problem for optical/magnetic recording channels modeled as linear systems, led to the discovery of continuous time neural networks ( Rama 5). The state updating scheme of the continuous time neural network takes the following form T

X(t) = Sign ( ∫ R(t , s) X( s) ds) 0

(2.23)

In this technical memorandum, the author for the first time associates energy functions with the state updating scheme. The multidimensional versions of these continuous time neural networks are discussed in (Rama 4).

Complex Neural Networks Neural networks in which the entries of the connection structure as well as state variables (indicating the binary states of the neuronal networks) are complex valued are already studied in one dimension. These results have the corresponding multidimensional versions. These results parallel the results for real neural networks. These results are aided by the fact that the quadratic form associated with a Hermitian symmetric matrix is always real and thus the eigenvalues of the Hermitian symmetric matrix are always real.

Adaptive Neural Networks These are neural networks in which the connection structure of the one/multidimensional neural network is varying with discrete/continuous time index. More explicitly, the connection tensor whose elements constitute the synaptic weights between the neurons that are located in one/two/multiple dimensions is varying with the time index in some orderly ( or random ) manner. The analysis of such one/multidimensional neural networks is being studied.

2.7 CONCLUSIONS A mathematical model of an arbitrary multidimensional neural network is described. This model is utilized to prove the convergence theorem for multidimensional neural networks. Utilizing the convergence theorem, multidimensional logic functions are defined and multidimensional logic synthesis is discussed. Infinite dimensional logic synthesis is briefly described. Various constrained static optimization problems of utility in control, communication, computation and other applications are summarized. Several innovative themes on one/multidimensional neural networks are summarized.

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Multidimensional Neural Networks: Unified Theory

REFERENCES (BoT) A. I. Borisenko and I. E. Tarapov, “Vector and Tensor Analysis with Applications,“ Dover Publications Inc., New York, (BrG) J.Bruck and J.W. Goodman, “A Generalized Convergence Theorem for Neural Networks”, IEEE Transactions on Information Theory, Vol. 34, No. 5, Sept 88. (CAB) S.T. Chakradhar, V.D. Aggarwal and M.L. Bushnell, “Neural Models and Algorithms for Digital Testing”, Kluwer Academic Publishers. (HoT) J. J. Hopfield and D. W. Tank, “Neural Computations of Decisions in Optimization Problems,“ Biological Cybernetics., Vol. 52, pp. 41-52, 1985. (Rama 1) Garimella Rama Murthy, “Multi/Infinite Dimensional Logic Synthesis,“ Manuscript in Preparation. (Rama 2) Garimella Rama Murthy, “Unified Theory of Control, Communication and Computation-Part 1,” Manuscript to be submitted to IEEE Proceedings. (Rama 3) Garimella Rama Murthy, “Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks: Constrained Static Optimization,” Proceedings of 2002 IEEE Information Theory Workshop, October 2002. (Rama 4) Garimella Rama Murthy, “Optimal Control, Codeword, Logic Function Tensors— Multidimensional Neural Networks, International Journal of Systemics, Cybernetics and Informatics, October 2006, pages 9-17. (Rama 5) Garimella Rama Murthy, “Signal Design for Magnetic and Optical Recording Channels: Spectra of Bounded Functions, “ Bellcore Technical Memorandum, TM-NWT-018026.

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CHAPTER

3

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks —Constrained Static Networks— Optimization

3.1. INTRODUCTION In the recent years, technological developments in parallel data transfer mechanisms led to HIPPI (high performance parallel interface), SMMDS (switched multi-megabit data service), FDDI (fiber distributed data interface). To match these high speed parallel data transfer mechanisms, multidimensional coding theory has been originated and some ad hoc procedures were developed for designing linear as well as non-linear codes. Multidimensional codes are utilized to encode arrays of symbols for transmission over a multidimensional communication channel. Thus, the central objective in multidimensional coding theory is to design codes that can correct many errors and whose encoding/decoding procedures are computationally efficient. A multidimensional error correcting code can be described by an energy landscape, with the peaks of the landscape being the codewords. The decoding of a corrupted codeword (array) which is a point in the energy landscape that is not a peak is equivalent to looking for the closest peak in the energy landscape. An alternative way to describe the problem is to design a constellation which consists of a set of points on a multidimensional lattice that are enclosed within a finite region, in such a way that a certain optimization constraint is satisfied. Neural network model, simulated annealing, relaxation techniques are some of the various computation models (based on optimization) that have been attracting much interest because they seem to have properties similar to those of biological and physical systems. The standard computation performed in a neural network is the optimization of the energy function. The state space of a neuro-dynamical system can be described by the topography defined by the energy function associated with the network. The connection structure of a neural network can either be distributed on a plane or in multidimensions (Rama 2). Thus, the field of multidimensional neural network theory and the field of multidimensional coding theory are linked through the common thread of optimization of

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Multidimensional Neural Networks: Unified Theory

multivariate polynomials (tensor based) over various subsets of the multidimensional lattice. In a nut shell, multidimensional error correcting codes and multidimensional neural networks can be associated with such polynomials. In contrast to the traditional ad hoc attempts to design multidimensional codes by a generation of researchers, the author for the first time discovered and formalized the idea of utilizing the theory of tensor spaces to represent and study multidimensional error correcting codes. The theory of tensor spaces enables the design of codes in one dimension (encoding as well as decoding techniques) to be translated to multi/infinite dimensions. Utilizing this representation, the author took a significant step forward in formally demonstrating the relationship between multidimensional neural networks, multidimensional codes and optimization of multivariate polynomials/monomials over various subsets of multidimensional lattice. This relationship provides new insights into the design of multidimensional encoders as well as decoders. Also, the relationships between concepts such as minimum distance, correctable errors of multidimensional codes can be derived through new proof arguments. Furthermore, the relationship enables the utilization of multidimensional decoding techniques for the solution of optimization of multivariate polynomials over the multidimensional hypercube ( other subsets of multidimensional lattice), a difficult problem that arises in various applied fields such as operations research, theoretical computer science etc. Also, utilizing the powerful techniques developed in these applied areas for such problems, new algorithms for maximum likelihood decoding of multidimensional error correcting codes can be designed. Thus, the results in this chapter are summarized in the following three paragraphs. The concepts of multidimensional neural networks, multidimensional error correcting codes, optimization of quadratic/higher degree forms based on components of a tensor (tensor component based multivariate polynomials), over various subsets of multidimensional lattice, are related from different viewpoints. It is proved that given a multidimensional linear block code, a neural network (generalized neural network) can be constructed in such a way that every local maximum of the energy function corresponds to a codeword tensor and every codeword tensor corresponds to a local maximum. It is shown that determining the global maximum of the energy function of a multidimensional neural network/generalized neural network is equivalent to performing the maximum likelihood decoding in a linear block multidimensional code. The results are generalized to multidimensional non-linear as well as non-binary codes. Theorems related to optimization of tensor based multivariate polynomials (terms/ monomials are based on the components of tensors) over arbitrary open/closed sets are proved. Infinite dimensional extension of the results is briefly discussed. This chapter is organized as follows. In section 2, after briefly reviewing the theory of multidimensional neural networks, it is proved that finding the global optimum of the energy function of the network is equivalent to finding a minimum cut in a certain

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

29

graphoid, the connection structure of a multidimensional neural network. In section 2, a connection between the multidimensional neural network model and graphoid based codes is established. It is shown that maximum likelihood decoding in a graphoid based code is equivalent to finding a minimum cut in a certain graphoid. Thus, it is shown that maximum likelihood decoding in a graphoid based code is equivalent to finding a maximum of the energy function in a multidimensional neural network. In section 3, the results are extended to general multidimensional linear block codes. A general energy function, not necessarily quadratic, is defined based on the generator tensor of a given linear block code. It is proved that finding the global maximum of the energy function is equivalent to maximum likelihood decoding in the code. In section 3, it is briefly discussed how the infinite dimensional codes are represented through the infinite order/dimension (either the order or the dimension or both is infinite) generator tensor (the entries of which satisfy some regularity conditions) and thus enable the infinite dimensional versions of the results to be derived. In section 4, the energy function associated with the parity check tensor of the multidimensional linear block code is described. When the tensor is written in the systematic form, it is shown that each codeword tensor corresponds to a local maximum of the multivariate polynomial associated with the parity check tensor and that each local maximum corresponds to a codeword tensor. The results are interpreted as the dual to the ones in the previous section for defining the Maximum Likelihood Decoding (MLD) problem. In section 5, the results are generalized to nonbinary codes. Further, in section 6, the results are generalized to non-linear multidimensional codes. In section 7, by means of a decomposition principle, theorems related to optimization of tensor based (based on the components of a tensor) multivariate polynomials over arbitrary open/closed sets are proved. Also, various innovative ideas on the utilization of results in previous sections, to derive very general results in static optimization are described. The chapter concludes with a summary of results derived. The results in this chapter are exactly the multidimensional versions of those in (BrB).

3.2 MULTIDIMENSIONAL NEURAL NETWORKS: MINIMUM CUT COMPUTATION IN THE CONNECTION STRUCTURE: GRAPHOID CODES A discrete time multidimensional neural network is a discrete time dynamical system represented by a weighted undirected connectionist structure in multidimensions. At each multidimensional neuronal element, there is a threshold value which will fire each neuron on crossing it. Each neuronal element computes an algebraic threshold function in the input variables. Let MN be a multidimensional neural network of dimension m and order n; then MN is uniquely specified by (S, T) where ( the number of neurons in each dimension is m i.e. the number of values assumed by each independent dimension variable) S is fully symmetric tensor of dimension m and order 2n, and T is a tensor of thresholds attached to neuronal elements with compatible order ( n ) and dimension ( m ). Every node can be in one of two

Multidimensional Neural Networks: Unified Theory

30

possible states +1 and –1. The state of node ( i1, i 2,..., in ) at time t is denoted by Xi 1, i 2..., in (t) . The state of MN at time t is the tensor X i1, i2,...1. in (t)of dimension m and order n. The state evolution at node ( i1, i 2,..., in ) is computed by

Xi 1, i 2,..., in (t + 1) = Sign (Hi 1, i 2,..., in (t ))

(3.1)

where

Hi 1,..., in (t ) =

m

m

j1= 1

jn = 1

∑ ... ∑ Si1,..., in; j1,..., jn X j1, j 2,..., jn (t) − Ti1,..., in (t)

The next state of the network i.e. X i1, i2,..., in (t + 1), is computed from the current state by performing the evaluation (3.1) at a subset of nodes of the multidimensional neural network, to be denoted by G. The modes of operation are determined by the method by which the subset G is selected in each time interval. If the computation (3.1) is performed at a single node in any time interval i.e. G| = 1 , then we will say that the network is operating in the serial mode, and if G|= mn , then we will say that the network is operating in the fully parallel mode. A state is called stable if and only if

Xi 1, i 2,..., in (t) = Sign (S ⊗ Xi 1,..., in (t) − Ti 1,..., in )

(3.2)

where ⊗ denotes inner product (the symbol is sometimes suppressed for notational brevity). Once a neural network reached such a state there is no change in the state of the network no matter what the mode of operation is. An important feature of the network MN is the convergence theorem stated below. Theorem 3.1: Let MN = (S, T) be a multidimensional neural network of dimension m and order n. S is a fully symmetric tensor of order 2n and dimension m . The network MN always converges to a stable state while operating in the serial mode (i.e. there are no cycles in the state space) and to a cycle of length utmost 2 while operating in the fully parallel mode.( i.e. cycles in the state space are of length utmost 2 ). This theorem is proved in (Rama 2). This theorem suggests the utilization of MN as a device for performing a local search of the optimum of an energy function. In the following, we formulate a problem that is equivalent to determining the global maximum of an energy function and how to map it onto a multidimensional neural network. Definition 3.1 Let G = (V, E) be a weighted and undirected non-planar graph in multidimensions where V denotes the set of nodes of G and E denotes the set of edges of G. Let K be the fully symmetric tensor whose components are the weights of the edges of G. Let V1 be a subset of V, and let V–1 = V–V1. The set of edges each of which is incident at a node inV1 and at a node in V–1 is called a cut in G.

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

31

Definition 3.2 The weight of a cut is the sum of its edge weights. A minimum cut (MC) of a non-planar graph/graphoid is a cut with minimum weight. In the following, we show the equivalence between the minimum cut problem in a graphoid (from now onwards, we call the connection structure of a multidimensional neural network also as a graphoid ) and the problem of maximizing the quadratic form as the energy function of a multidimensional neural network. Every non-planar graph including the connection structure of a multidimensional neural network is a Graphoid (by definition). Theorem 3.2: Let MN = (S, T) be a multidimensional neural network with all the thresholds being zero i.e. T = 0. The problem of finding a state V for which the quadratic energy function E is maximum is equivalent to finding a minimum cut in the graphoid corresponding to MN. Proof: Since T = 0, the energy function is given by

E=

m

m

m

m

∑ ... ∑ ∑ ... ∑ S

i1= 1

in = 1 j 1 = 1

jn = 1

i 1,..., in ; j 1,..., in

Xi 1,..., in

X j 1,..., jn

(3.3)

Let (i1,i2,...,in) = i,(j1,...,jn) = j. Let S + + denote the sum of weights of edges in MN with both the end points being in the same vertex set of the cut i.e. i=j =1, and let S − − ; S + − denote the corresponding sums of the other two cases. It follows that E = 2 ( S++ + S− − − S+− ) which can also be written as E = 2 ( S++ + S− − + S+− ) – 4 S+ –

(3.4)

Since, the first term in the above equation is constant (it is the sum of weights of the edges), it follows that the maximization of E is equivalent to the minimization of S+–. Clearly, S+– is the weight of the cut in MN with V1 being the nodes of MN with a state equal to 1. Q. E. D.

Graphoid Based Codes In this sub section, relationship between multidimensional neural networks and error correcting codes based on graphoids is investigated. The ‘multidimensional error correcting codes’ associated with graphoids (connection structure of a multidimensional neural network ), are called “graphoid-theoretic” codes. The family of graphoid codes are defined based on the tensors naturally associated with the connection structure of a multidimensional neural network with nodes as well as edges. Let G = (V, E) be an undirected connectionist structure of a multidimensional neural network with weights on the edges. Like a graph in the plane, this is a representation for a

Multidimensional Neural Networks: Unified Theory

32

non-planar graph type structure called graphoid (not necessarily the connection structure of a multidimensional neural network). Consider a fully symmetric tensor of dimension m and order 2n; which is utilized to describe the connection structure of a multidimensional neural network. A subset of the set of edges of G can be represented by a characteristic tensor of order 2n with the edge between two nodes Vi 1, i 2,..., in , Vj 1, j 2..., jn , leading to an entry of +1 at those locations in the tensor. Thus, an edge characteristic tensor of a graphoid E is defined such that

Eˆ i1,..., in; j1,..., jn

 1 if an edge is incident between  =  nodes(i1,...., in )and ( j1,...., jn ).  0 otherwise. 

(3.5)

Definition 3.3 The incidence tensor of a graphoid G = (V , E ) is a block tensor of the form

TVˆ1    TVˆ2  DGˆ = .  ..  (3.6)   TVˆ   n   where TVˆ represents the tensor of the set of edges incident upon the node Vi . It should be noted that the incidence tensor is a blocked tensor and the above illustration is shown to aid the imagination of the reader. Various concepts associated with planar graphs are utilized as the basis to define the following concepts associated with a graphoid (non-planar). They provide the notation associated with graphoid theoretic codes. The following lemmas are very easy to verify. Lemma 3.1: The set of characteristic tensors that correspond to the cuts in a connection i

structure G = (V , E ) of a multidimensional neural network form a linear tensor/m-d vector (depending on the notational convenience) space over GF(2) in multidimensions of dimension

( V − 1)

.

The linear tensor/m-d vector space that corresponds to the cuts of a graphoid Gˆ will be called the only cut space of Gˆ . Furthermore, the circuits in a graphoid also constitute a linear tensor/vector space. Lemma 3.2: Given a connected graphoid G = (V , E ) ; the incidence tensor of Gˆ has rank

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks ( V − 1)

33

. Every block tensor in DGˆ associated with a node is a characteristic tensor of a cut

(

)

and every Vˆ − 1 block tensors of DGˆ corresponding to different vertices/nodes of the graphoid form a basis for the cut space of Gˆ .

Hence, given a connection structure Gˆ , the cut space of the graphoid is a

(

)

multidimensional linear block code of dimension Vˆ − 1 . For the sake of brevity, in the following, we only consider ‘cut codes’. Given a graphoid, Gˆ , an interesting question is how to formulate the maximum likelihood decoding (MLD) problem of the code CGˆ in a graphoid-theoretic language.

(

)

That is, given a graphoid Gˆ = Vˆ , Eˆ and a (0, 1) tensor Y of dimension m and order 2n; what is the codeword in C ˆ closest to Y in Hamming distance? G The following lemmas will answer the questions. Hamming Distance: Given two (0,1) tensors, X, Y; the Hamming distance between m dimensional tensors of order 2n is the number of places where they differ. This definition is motivated by transmitting a binary tensor X through a noisy multidimensional channel, observing the output Y and counting the number of errors that have occurred.

(

)

Lemma 3.3: Let Gˆ = Vˆ , Eˆ be a graphoid. Let CGˆ be the multidimensional code associated with Gˆ . Let Y be a (0,1) tensor of order 2n (dimension m). Construct a new graphoid, to be defined/denoted by Gˆ Y; by assigning weights to the edges of Gˆ as follows: Wi1, i2,..., in; j1,..., jn = (–1) Yi1,..., in; j1,..., jn ((−1)Power…) (3.7)

Wi 1, i 2,..., in ; j 1,..., jn is the weight associated with the edge (i1,..., in ; j1,..., jn ) in Gˆ . Then the maximum likelihood decoding of the tensor Y with respect to CGˆ is equivalent to finding the minimum cut in Gˆ Y . Proof: Assume the number of ones in Y is b. Let P be an arbitrary codeword in CG. Let L i,j denote the number of positions in which P contains an i ∈ {0, 1} and Y contains a j ∈ {0,1}. Clearly,

b = L0,1 + L1,1

(3.8)

−L1,1 + L1, 0 = L0,1 − b + L1,0

(3.9)

Thus,

= L0,1 + L1,0 − b

(3.10)

Multidimensional Neural Networks: Unified Theory

34

Minimizing the right hand side of the above expression over all P ∈ C G is equivalent to finding a codeword which is the closest to Y. On the other hand, minimizing the left hand Q.E.D side is equivalent to finding the minimum cut in G Y. From the above lemma, the following theorem follows.

(

)

Theorem 3.3: Let Gˆ = Vˆ , Eˆ be a graphoid. Then, maximum likelihood decoding of a tensor word Y with respect to CGˆ is equivalent to finding the maximum of the quadratic energy function E of the multidimensional neural network defined by the graphoid Gˆ Y with all its threshold values equal to zero. Proof: By Lemma 3.3, maximum likelihood decoding of Y with respect to C Gˆ is equivalent to finding the minimum cut in Gˆ Y . By Theorem 3.2, finding the minimum cut in a graphoid is equivalent to finding the global maximum of the energy function (quadratic) of a multidimensional neural network defined by a graphoid with all the thresholds at each neuronal element set to zero. Q.E.D. Graphoid based error correcting codes are very limited since the connection structure of a multidimensional neural network is represented by a fully symmetric tensor. This imposes restrictions on the minimum distance of multidimensional codes. Thus, a natural question that arises is whether the equivalence stated above in the Theorem 3.3 can be generalized to arbitrary multidimensional linear block codes. Graphoid codes arose naturally out of the topological properties of the connection structure of a multidimensional neural network. The connection structure required a fully symmetric tensor to represent it. The neural network model enabled the association of a quadratic energy function with the fully symmetric tensor and its optimization over the multidimensional hypercube. Thus, the encoders and decoders of graphoid codes are defined through topological structure and optimization of multivariate polynomials. Since, an arbitrary tensor like the fully symmetric tensor constitutes a linear operator, unlike graphoid codes, arbitrary multidimensional linear codes are first defined through their algebraic structure in the next section. Then the maximum likelihood decoding problem of such codes is discussed.

3.3 MULTIDIMENSIONAL ERROR CORRECTING CODES: ASSOCIATED ENERGY FUNCTIONS—GENERALIZED NEURAL NETWORKS Recent advances in high speed parallel data transfer mechanisms based on light wave/ optical networks motivated the design and analysis of multidimensional codes. Several researchers utilized ad hoc techniques (sometimes pseudo-mathematical techniques) to design and analyze multidimensional codes based on the extensions of the ideas in one dimensional error control coding theory. The author for the first time developed the idea of utilizing ‘tensor linear operator’ for the design and analysis of multi/infinite dimensional linear as well as non-linear codes conceived as sub-spaces over tensor spaces.

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

35

A multidimensional linear codeword constellation is defined as the subspace of a tensor linear space defined through the generator tensor. The encoding operation of a multidimensional (m, n ; m,l) linear code, defined by an m dimensional (n+l) order generator tensor Gi 1, i 2,..., in ; j 1, j 2,..., jl is performed in the following manner: An m dimensional information tensor of order n, Bi1, i2,..., in (with 0, 1 symbols) is encoded into the m-dimensional “codeword tensor”, C j 1, j 2,..., jl (constellation member) of order ‘l’ by the following tensor inner product (outer product followed by contraction over appropriate indices) scheme:

Bi 1, i 2,...., in ⊗ Gi 1, i 2,...., in ; j 1, j 2,..., jl

= C j 1, j 2,..., jl

(3.11)

where ⊗ denotes the inner product operation (between tensors defined over a finite field) by means of exclusive or operation between the components of outer product of tensors (contraction over appropriate indices of the sum of products of binary variables). The above procedure of generating the codeword tensor from an information tensor leads to the following interesting considerations which are inherent to multidimensional code design. In one dimension, a binary information vector of length k is encoded into a codeword vector of length n by padding the parity bits to it. The parity check equations obtained through the parity check matrix determine these bits. In the case of two/multidimensional array of information bits, there are many ways to encode the array into a codeword array. Even in the simplest two dimensional array case, by padding a border of parity bits along the row wise as well as column wise directions, the codeword array can be generated. In the following, this degree of freedom in multidimensional coding is formally described. A multidimensional information array (information tensor) is mapped into a codeword array in the following ways: (1) An m-dimensional information tensor of order n is mapped into an m-dimensional codeword tensor of order l (l > n), (2) An m -dimensional information tensor of order n is mapped into k -dimensional codeword tensor (k > m ) of order n, (3) An m-dimensional information tensor of order n is mapped into a k -dimensional (k > m) codeword tensor of order l (l > n). For the purpose of notational convenience, in the following encoding through the operation (1) is only utilized. It is easy to realize that by transposing the information as well as generator tensors, the operation (2) in encoding is achieved. But to encode an information tensor into a generator tensor through the operation (3), a second generator type tensor is utilized. Various ideas familiar in one dimensional coding theory (parity check matrices, primitive polynomials, basis, cosets etc.) have corresponding parallels in multi/infinite dimensional coding theory based on the tensor linear operator defined over a finite field. The detailed translation from one dimensional encoding/decoding algorithms to

36

Multidimensional Neural Networks: Unified Theory

multidimensional encoding/decoding algorithms is done by utilizing tensor algebra concepts with parallel linear algebra concepts.

Infinite Dimensional Codes Now let us consider infinite dimensional codes. An infinite dimensional tensor can be of the following types: (a) the dimension of the tensor is finite, whereas the order is infinite, (b) the dimension of the tensor is infinite, whereas the order is finite, (c) the dimension as well as order of the tensor are infinite. An infinite dimensional code can be generated in the following manner. It is assumed that the generator tensor of the code is such that either the dimension or the order or both are infinite. Also, it is assumed that the entries of the generator/information tensor satisfy the regularity conditions necessary to ensure that the inner product makes sense (convergence of the partial sums of outer product to a limit etc.). (i) An information tensor of finite dimension/order is mapped into a codeword tensor of infinite dimension/order. This type of encoding can happen in practical multidimensional communication systems, (ii) An infinite dimension/order ( or both are infinite) tensor is mapped into a codeword tensor with either the dimension or the order or both being infinity. In the above encoding schemes, the generator/parity check tensors are of compatible dimension/order (with the information tensor being encoded) to ensure that a proper infinite dimensional codeword is generated. Infinite dimensional extensions of the results in sections 3, 4, 5, 6, 7 (to be described in the following paragraphs) follow from the immediate extensions of the formal arguments to infinite dimensional tensors that satisfy the regularity conditions. They are not explicitly repeated. In the following, a very brief summary of multidimensional information theory is provided as it is based on the tensor linear space structure idea necessary to model multidimensional arrays.

Multi/Infinite Dimensional Information Theory In one dimension, a mathematical theory of communication is developed utilizing the concepts of information/entropy associated with a random variable, conditional entropy, joint entropy etc. These concepts are the vital tools to prove the noiseless channel coding theorem. Various channel models are developed. The concepts of mutual information, capacity of a discrete memoryless channel are utilized to prove the second channel coding theorem. One dimensional information theory then led to rate distortion theory. In multidimensions, a source generates multidimensional arrays of information which pass through a multidimensional channel. A multidimensional independent, identically distributed information array of symbols is associated with the concept of entropy H (Xi1, i2,..., in) in the following manner:

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

H (Xi 1, i 2,..., in ) =

m

m

i1= 1

in = 1

∑ ...∑ Pi1, i 2,..., in log (1 Pi1, i 2,..., in )

37

(3.12)

Given the basic idea of the above definition, results from one dimension are generalized to multidimensions utilizing the principles described in (Rama 3). Complex sources such as a Markovian Source require some sophistication in defining the entropy/uncertainty of the source. The interesting channel model in multidimensions is the discrete memoryless channel represented through a stochastic tensor whose elements are conditional probabilities Pj 1,..., jl , i 1,..., ik . This corresponds to a Markov random field. Detailed theorems are derived utilizing the principles described in (Rama 3). With the multidimensional encoding scheme formally described, it is proved in the following that the maximum likelihood decoding problem of a multidimensional linear block code is equivalent to the maximization of multivariate polynomial (whose terms/ monomials are described in terms of the entries of received, generator tensors) associated with the generator/received tensors over the multidimensional hypercube. The essential idea in the derivation of the desired result is (generalization of Theorem 3.3 to arbitrary multidimensional linear codes) to represent the symbols of the additive group as symbols in the multiplicative group through the following transformation: (3.13)

a → ( − 1) a i .e . 0 → 1, 1 → − 1 .

Thus, the information tensor Bi 1,..., in is represented by the tensor Xi 1,..., in , where the component Xi 1,..., in =

(–1)

Bi 1,...in

. The encoded codeword C j 1,..., jl is thus represented by the

tensor Yj 1,..., jl . Hence, a component of the tensor Y is given by C j1 ,...., jl

Yj1, j2 ,.... Jl = ( −1)

m

m

i1 = 1

in = 1

i1,.....,in = ∏ ...∏ X i 1,...., in

G

; j1,...... jl

(3.14)

Definition 3.4 In the {1, –1} representation of a multidimensional linear code, instead of a generator tensor, given an information tensor Xi 1,..., in , an encoding procedure X → Y is utilized, where the tensor Y j1,..., jl is such that Y j1,..., jl component is a monomial that consists of a subset of the

X i1,..., in . An encoding procedure is systematic if and only if Y j1,..., js = X j1,..., js for 1 < s < n. Definition 3.5 Let Gi1, i 2,..., in ; j1, j 2,...., jl be a generator tensor of ones and zeroes. The polynomial representation of generator tensor G with respect to a {+1, –1} received tensor of dimension m and order l, W denoted by E is,

Multidimensional Neural Networks: Unified Theory

38 m

m

i1 = 1

in =1

EW ( X ) =W ⊗ ∏...∏ X i1,...,in = W ⊗Y (X )

(3.15) (3.16)

where ⊗ denotes inner product between the tensors (i.e. outer product of the tensors followed by contraction over appropriate indices). Consider the linear multidimensional block code defined by the generator tensor G (or equivalently by the encoding procedure associated with G ). The polynomial representation of G i.e. EW ( X ), will be called the energy function of W with respect to the encoding procedure X → Y . To establish the connection between the energy functions (optimized by neural/ generalized neural networks over various subsets of the multidimensional lattice) and linear multidimensional block codes, we will prove that finding the global maximum of EW (X) is equivalent to maximum likelihood decoding of a tensor W with respect to the code C. Theorem 3.4: Given an ( m, l ; m, n ) multidimensional linear block code C defined by an encoding procedure X → Y , and a tensor W of ones and minus ones i.e. a {+1, –1} tensor, the closest codeword (in Hamming distance) to W in C corresponds to an information tensor B if and only if EW (B) = Maximum overall tensors X of {EW(X)}. (3.17) Proof: For an { +1, –1 } information tensor,X the scalar energy function is given by EW (X) = W ⊗ Y(X)

(3.18)

= {( j1, j 2,..., jl ): W j1,..., jl = Y j1,..., jl ( X )} − {( j1,..., jl ): W j1,...., jl ≠ Y j1,...., jl ( X )}

(3.19)

= ml − 2 {( j 1,... jl ): Wj 1,..., jl ≠ Yj 1,..., jl (X )} = ml − 2 dH (W , Y)

(3.20)

where d H denotes the Hamming distance between the multidimensional codewords W, Y . From the above expression, EW ( B ) will achieve a maximum if and only if d H (W , Y) achieves a minimum. Q. E. D.

Minimum Distance of Linear Multidimensional Block Codes Given an encoding procedure, we can use the same argument as in the above theorem, to express the minimum distance of the code. Consider the encoding procedure:

X = (Xi 1,..., in ) → Y = (Yj 1,..., jl ) and the energy function with W, a tensor with all the components equal to one.

(3.21)

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

EW (X ) =

m

m

j1= 1

jl = 1

∑ ...∑ Yj1 ,..., jl

39

(3.22)

i.e. sum of all the components of tensor Y. As in Theorem 3.4,

EW ( X ) = m l − 2d H ((all ones tensor), Y )

(3.23)

and minimum over all X tensors ≠ (all ones tensor) of d H ((all ones tensor), Y) occurs at M =Maximum overall tensors other than all ones tensor of EW (X )

(3.24)

Thus, d * ( the minimum distance of the code is given by ) d* =

ml – M 2

(3.25)

The above results are being generalized to infinite dimensional codes utilizing infinite dimension/order tensors . Game-Theoretic Codes: Optimal Codes In the theory of error correcting codes, minimum distance of a linear code provides a measure of the number of errors that can be corrected. From ( 3.25), it is evident that the maximization of minimum distance of a multidimensional linear block code requires minimizing M. Thus, we have the following Lemma. Lemma 3.4: The multidimensional (m, n ; m, l) linear block code which minimizes M in (3.24) enables the correction of maximum number of errors among all possible such error correcting codes: Proof: From (3.25), maximization of minimum distance of an (m, n ; m , l) linear code is equivalent to minimizing M, i.e. minimizing the maximum value of the energy function over the m-d hypercube ( excluding the all ones tensor). Such a code design problem fits in the game-theoretic framework. It is well known that maximization of minimum distance also maximizes the number of errors that can be corrected. Q. E. D.

3.4 MULTIDIMENSIONAL ERROR CORRECTING CODES: RELATIONSHIP TO STABLE STATES OF ENERGY FUNCTIONS Let C be a linear multidimensional block code (over GF(2)) defined by the generator tensor G. Let EC be a polynomial over the components of {+1, –1} tensors (energy function) with the property that every local maximum in EC corresponds to a codeword in C and every codeword in C corresponds to a local maximum in EC. An interesting problem is,

Multidimensional Neural Networks: Unified Theory

40

given a code C defined by G, the generator tensor, is there an efficient algorithm to construct to EC ? In the following, the above problem is solved by considering the parity check tensor of a multidimensional linear block code. Consider an (m, l ; m , n) linear multidimensional block code. Without loss of generality, let us consider the generator tensor G given in the systematic form i.e.

Gj 1, j 2,..., jn = I j 1,..., jl Pjl + 1,..., jn

(3.26)

where I j 1,..., jl is an identity tensor of compatible order. The parity check tensor of C is denoted by H and is given by

P  HT = H j ( n − 1),..., ,..., j 1 =  I  (3.27)   i.e. a blocked tensor with sub-tensors of compatible dimension and order. From the definition of a parity check tensor of a multidimensional linear block code, C j 1,..., jn ⊗ H j ( n −1),..., ,..., j 1 = 0

(3.28)

where the multidimensional tensor codeword on appropriate/compatible inner product (outer product followed by contraction over the appropriate indices) with the parity check tensor gives the zero tensor. The above equation can be rewritten using the polynomial representation of generator tensor devised in the previous section (with the tensor of coefficients being the all-ones tensor. It should be noted that the all-ones tensor in the {1, –1} representation corresponds to all-zero tensor in the {0,1} representation). Lemma 3.5: Let E (X) be the polynomial representation of parity check tensor HT with respect to the all ones tensor. Then, X ∈ C , the multidimensional linear block code if and only if

E (X) = m(n–l). Proof: E , the polynomial representation of parity check tensor has m (n–l) terms, and all the coefficients are equal to one. Hence, E = m(n–l). if and only if all the terms are equal to one. Q. E. D. The above Lemma ensures that in the polynomial representation, E (X), every codeword corresponds to a global maximum (stable state). An interesting question is, does every local maximum correspond to a codeword. This question is answered by the following theorem. Theorem 3.5: Let C be a linear multidimensional block code, with G, H, EC, and E as defined above. Then E is a polynomial with the properties of EC. That is, X corresponds to ∈C. a local maximum in E if and only if X∈

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

41

Proof: From the above Lemma, the global maximum of E is m (n–l) ; thus every codeword is a global ( and thus a local ) maximum. The converse follows from the fact that the tensor H has a systematic form. Specifically, the last m (n–l) variables in E ; i.e., xi1,i 2 ,...., in – l +1 ,...., in ; where the order indices iˆn −l +1 ,..., iˆn (each of them) assume m values, each appear only in

one term. That is, since I is an identity tensor in the parity tensor H; x i 1,..., iˆn – l +1 ,...., iˆn appears only in first term, and so on. Now, assume that a tensor V exists that corresponds to a local

(n−l ) maximum (which is not global maximum). That is E (V) = L, where L < m . Hence, at least one term exists in E (V ) that is not one. However, this can be made one by flipping the value of the index variables that appear in this term. This contradicts the fact that V is a local maximum. Q. E. D. To summarize, given a linear code C, the algorithm for constructing a polynomial is as follows:

(1) Construct the systematic generator tensor of C by the standard techniques in tensor algebra, (2) Construct the systematic parity check tensor of C in accordance with (3.27) (3) Construct E , which is the polynomial representation of H with respect to the allones tensor. By the above Theorem 3.5, EC = E . In the following, generalizations of the above results are discussed. Also, some important comments, remarks are provided. (A) The construction just described also works for cosets of linear multidimensional block codes. Let W be a tensor of dimension m and order (n – l) of the coefficients of E. In the construction described above, the all-ones coefficient tensor was chosen and it was concluded that EC = E . It corresponds to the all-zero syndrome tensor. Let C be a coset of C, and let T be the syndrome which corresponds to C. Utilizing the proof argument of Theorem 3.5, it can be proven that a one-toone correspondence exists between the local maxima of polynomial representation of the parity check tensor H with W = T and the tensors in the coset C. Clearly, the syndrome that corresponds to the code C is the all-ones tensor (by noting that in the transformation in section 3, 0 goes to 1). (B) The construction described in this section is a dual way of defining the maximum likelihood problem (MLD) (with respect to the one suggested in section (3)). Consider a linear multidimensional block code defined by the parity check tensor H. Given a tensor V, the maximum likelihood decoding (MLD) problem can be defined as finding the local maximum in EC closest to V or, equivalently, finding a local maximum of the energy function associated with the syndrome (corresponding to V) that is achieved by a tensor of minimum weight. The above results are generalized to some infinite dimension/order tensors in a straightforward manner. In the following section, the above results are generalized to non-binary codes.

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42

3.5. NON-BINARY LINEAR CODES In the following, results on maximum likelihood decoding of non-binary linear codes are discussed. Consider a linear multidimensional block code over a finite field GF(p) with p being a prime. For the sake of notational convenience, we first consider an (m, k ; m, n) linear multidimensional block code which maps a transmitted input tensor of dimension m, order k into a codeword tensor of dimension m and order n. Let G denote the generator tensor of the code which maps the ( m, k ) input tensor into a (m, n ) codeword tensor. Then, m k symbols of the input tensor B in Zp are encoded into the codeword V by the procedure:

Vi 1, i 2,..., in = (Bi 1, i 2,..., ik ⊗ Gi 1,..., ik ; j 1,..., jn ) mod p

(3.29)

The essential idea is once again to utilize the multiplicative representation. Let u be the p th root of unity i.e. µ = e ( j 2 Π )/ p (3.30) The additive Zp group can be represented as a multiplicative group of p th roots of unity through the transformation: a → u a In the multiplicative representation, the information symbols in information tensor are represented as X i 1,..., ik = u Bi 1,....,ik

(3.31)

Thus, the encoded codeword tensor V, is represented by a new tensor Y, where

Yi 1,..., in = u

Vi1 ,..., in

= u

m m .. . ∑ ( B i 1 ,. .. , ik ∑ i 1 = 1 ik = 1

m

m

i1= 1

ik = 1

m

m

i1= 1

ik = 1

= ∏ ...∏ u

G i 1 ,. .. , ik ; j 1 ,. .. , j n ) m o d p

(3.32)

Bi1 ,..., ik Gi1 ,...,ik ; j1 ,..., jn

= ∏ ...∏ Xi1,...,i 1,...,ik ik ; j 1,..., jn G

Hence, as in the case of a binary linear multidimensional code, we can represent a multidimensional code over a field (finite) with p elements (p is a prime) by an encoding procedure. The elements are now p th roots of unity. Thus, given an information tensor

X = (Xi 1,..., ik ) , we have the one-to-one assignment X = (Xi1,.....,i k)

–→

Y = (Yi1,....in)

(3.33)

where Y = (yi 1,..., in ), is a monomial. We discuss the maximum likelihood decoding problem with respect to two different distance measures. In the first generalization, we consider solving the maximum likelihood

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43

decoding (MLD) problem with the metric being the Hamming distance between the tensors while in the second case, we consider the Lee distance. The generalization for the case where the Hamming distance is utilized in the maximum likelihood decoding (MLD) problem is based on the following well known Lemma. Lemma 3.6: Let p be a prime, and let ( j 2 π

p ). Assume KE{0,1,2,...,(p − 1)}µ=e then

1, if k = 0, = m =0 0, othe rw ise The generalization is stated through the following theorem. (1 p )

( p −1)

∑u

km

(3.34)

Theorem 3.6: Consider an (m, k ; m, n) multidimensional linear block code over GF(p), with p

p being a prime. Let X → Y be the corresponding encoding procedure. Let EW be the following multivariate polynomial representation of the generator tensor G with respect to an arbitrary received tensor W : p

EW (Y ) =

( p −1)

∑ (W l=0

• i 1,..., in

⊗ Yi 1,..., in )

(3.35)

where W• denotes the complex conjugate of W and ⊗ denotes the inner product between the tensors. Then, the maximum likelihood decoding of W i1 ,......, i n is equivalent to finding p

the maximum of EW (Y ) . Proof: It follows by the same argument as Theorem (3.4) adopted to the variables appearing p

in the polynomial EW (Y) and the application of above Lemma.

Q.E.D.

The essence of the above theorem stated in more explicit language leads to the following conclusion. Given a received tensor Wi 1,..., in , the closest codeword tensor (in Hamming distance) to W in C (the code utilized at the input to the multidimensional channel) corresponds to a tensor B if and only if ( p −1) Max l EW (B ) = All tensors EW (Y ) = ∑ (W i 1,..., in ⊗ Yi 1,..., in ) l =0

(3.36)

Next, we consider the maximum likelihood decoding problem with respect to the Lee distance. We first consider the cases where p = 3 or 5. In these cases, there are easy expressions for the energy function. It is convenient to redefine the energy function in the following manner: Given an encoding procedure for a transmitted tensor X = (Xi1,..., i k), into a codeword tensor Y = (Yi1,..., in), by the following procedure i.e.

X = (Xi 1,..., ik ), → Y = (Yi 1,..., in )

(3.37)

Multidimensional Neural Networks: Unified Theory

44

and W = (W i1,..., in), a tensor whose entries are the pth roots of unity, we redefine the energy function as follows: i EW ( X ) = Re(W i1 ,....,in ⊗Yi1 ,..., in )  (3.38) where Re (x) denotes the real part of the complex number,  x  denotes the integral part of the number x and xi denotes the complex conjugate of x. It should be noted that the energy function coincides with the one for p = 2 (in the case u = –1). The definition of Lee distance is provided to facilitate the easier understanding of further discussion. Definition 3.6 p The Lee weight of an m-dimensional tensor of order k , X = (Xi 1,..., ik ), (Xi 1,..., ik ) ∈ Z , p is a

prime, is defined as m

m

i1=1

ik =1

WL = ∑ ...∑ Xi 1,..., ik

(3.39)

where 0 < X i 1, i 2 ,..., ik ≤ ( p 2)  X i 1, i 2,..., ik , X i 1, i 2,..., ik =   p − X i 1, i 2 ,..., ik , ( p 2) < X 1i , i 2,..., ik < ( p − 1) The Lee distance between any two compatible tensors is defined as the Lee weight, W L of their difference.

With the above definition, we study the cases where p = 3, p = 5. From now, in the following discussion, X → Y denotes the encoding procedure that defines a code (multidimensional), and X , Y are tensors of dimension m and order k , n respectively, of third or fifth roots of unity. In the following, two new theorems are proved. The first one is equivalent to the Theorem (3.4). It states that maximum likelihood decoding (MLD) in a ternary code is equivalent to the maximization of the energy function in (3.39). The Theorem is formally stated below: Theorem 3.7: Let p = 3, A → B; then B is the closest multidimensional codeword (in the Hamming distance) to a received tensor word W if and only if

EW ( A ) = Max EW (X ). X

(3.40)

Proof: The proof is similar to that of Theorem (3.4) and is avoided for brevity Q.E.D. The proof of Theorem (3.7) as well as Theorem (3.8) requires the utilization of Lemma (3.6) and a clear understanding of when the energy function is maximized. The new

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

45

energy function provided in (3.39) is a convenient expression. Its definition is once again based on understanding when the energy function is maximized and how utilizing Re ( ), the real part of a complex number does not alter the end effect in decoding a received word. Now, we consider the problem of maximum likelihood decoding (MLD) with respect to the Lee distance: Theorem 3.8: Let p = 5, A → B ; then B is the closest multidimensional codeword (in Lee distance) to a received tensor word W if and only if

EW ( A ) = Max EW (X ). X

(3.41)

Proof: From the definition of the energy function in i EW (A ) = (i 1, i 2,..., in ) : W ( i 1,..., in ) ⊗ B i 1,..., in = 1

{

}

{

i − (i 1, i 2,..., in ) : W( i 1,..., in ) ⊗ B i 1,..., in = u 2 or u 3

}

(3.42)

i = m n − (i 1, i 2,..., in ) : W ( i 1,..., in ) ⊗ B i 1,...., in = u or u 4

{

{

− 2 (i 1, i 2,..., in ) : W i ( i 2,..., in ) ⊗ Bi 1,..., in = u 2 or u 3 = mn − dL (W,B)

}

} (3.43)

Where W• denotes the complex conjugate of all components of W and d L denotes the Lee distance. Hence, EW (A) reaches a maximum if and only if d L (W, B) reaches the minimum. Q.E.D. The above results are generalized to infinite dimension/order tensors in a straightforward manner.

3.6 NON-LINEAR CODES In the theory of error control codes in one dimension, linear block codes are first extensively studied and various problems including the sphere packing problem was subjected to intense theoretical investigations. The research and development led to various theoretical as well as practical encoding/ decoding algorithms. Then, because it was thought that linear codes are limited from the point of view of various code parameters such as the number of (Ara) correctable errors/minimum distance, non-linear block codes were studied. The research in this direction culminated in the discovery of codes from algebraic geometry based techniques. The encoding algorithm was generally easy from the point of view of theory as well as physical hardware. It is the decoding algorithm which was considered difficult and was the subject of intense investigations resulting in several decoders. The maximum likelihood decoding (MLD) problem of linear codes and the relationship to energy functions (discussed

Multidimensional Neural Networks: Unified Theory

46

in the previous sections) naturally suggests a search for similar techniques to non-linear codes. In the following, non-linear multidimensional codes are investigated. The essential idea in generalizing the results in previous section to non-linear multidimensional codes is to consider the representation of Boolean functions as polynomials over the field of real numbers. In the context of one dimensional non-linear codes, part of the discussion is known (BrB) and is repeated here for the sake of completeness. Also, utilization of some subtle ideas associated with tensor products make the presentation essential aid for realizing that non-linear multidimensional codes share various features with linear codes. Definition 3.7 A Boolean function f on n variables, is a mapping

f : {0,1}n → {0,1}

(3.44)

For the present discussion, it is useful to define Boolean functions using the symbols 1 and –1 instead of the symbols 0 and 1, respectively. Definition 3.8 A Hadamard matrix of order m, denoted by H , is an m × m matrix of +1’s and –1’s such that

Hm HmT = mI m ,

(3.45)

where Im is the m × m identity matrix. The above definition is equivalent to the assertion that any two rows of H are orthogonal. Hadamard matrices of order 2k exist for all k > 0. The construction is as follows: H1 = [1]

1 1  H2 = 1 – 1   

 H 2n H 2n  . H 2n +1 =   H 2n − H 2n 

(3.46)

Definition 3.9 Given a Boolean function f of order n, P is a polynomial (with the coefficients over the field of real numbers) equivalent to f if and only if for all vectors X ∈ {1, – 1}

n

f (X ) = Pf (X ).

(3.47)

An important problem that is relevant to the investigation of non-linear multidimensional codes is the following: Given a Boolean function f of order n, compute Pf , polynomial which is equivalent to f. From the results in section 3, it is evident that the components of the codeword tensor (of a linear code), in the {1, – 1} representation are Boolean functions (monomials) in the

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

47

components/elements of the information tensor through the mapping described in Definition 3.4. Thus, the idea once again is to represent the corresponding Boolean functions of non-linear codes as polynomials/monomials over the field of real numbers. In the context of vector variable, the following inferences from Theorem (3.4) are well known in switching theory textbooks. But, in the case of vector variables, in (BrB), an alternative proof is given. In the following, it is shown that given the Boolean functions which are the components of a codeword tensor, there exist polynomials (with coefficients over the field of real numbers) equivalent to them over the multidimensional hypercube. Theorem 3.9: Let f be a Boolean function of order either strictly less than or equal to m n (in the components of a tensor X of dimension m and order n). Let Pf be a polynomial equivalent n to f. Let B denote the tensor of coefficients of Pf . Let P denote the tensor of utmost 2 m values of Pf (corresponding to m n{+1, – 1} components of tensor X ). Then, (1) the polynomial Pf always exists and is unique, (2) the following relationship is satisfied P = G ⊗ B, where ⊗ denotes the inner product of tensors. Proof: The proof is constructive in nature. The essential idea is to determine the coefficients of the polynomial by solving a system of linear equations, possibly imbedded in tensors. First, let us consider a Boolean function f of one variable and let us determine the coefficients of the polynomial Pf . Pf (x) = b 0 +b 1 x

(3.48)

Evaluating the polynomial on the domain of the Boolean function, we have

Thus,

Pf (1) = b 0 + b 1

(3.49)

Pf (–1) = b 0 – b 1

(3.50)

P = G ⊗ B,

 +1 + 1 where G =  +1 – 1   

(3.51)

G is a Hadamard matrix and B as defined before is the vector of coefficients of

Pf (X1 , X2 ,..., Xn + 1 ) . Remark Before proceeding with the proof, the following comparison/discussion on the similarities and differences between tensor products, matrix products is very relevant. Consider a G matrix and a column vector B. The tensor product, when the variables (matrix, column vector) are treated as tensors is given by

Multidimensional Neural Networks: Unified Theory

48

CONTRACTION → Pi G ⊗ B = Gi , j Bk = Pijk 

G11 B1 G21 B1

G11 B2 G12 B1 G21 B2 G22 B1

G12 B2 G22 B2

(3.52)

Now, we perform contraction on certain indices of the tensors. The resulting tensor is a first order tensor. Specifically, suppose we do the contraction over the indices j, k. Then, we have

G11 B1 + G12 B2 G21 B1 + G22 B2

(3.53)

Thus, the tensor product, in contrast to the matrix product allows more freedom in summing the components over different indices (contraction over different indices in the language of tensor algebra) of the tensor. Now, we return to the original proof. The above argument is now generalized to less than or equal to m n variables ( or arbitrary finite/countable number of variables which are possibly the components of a tensor ) by the method of mathematical induction. The case m = 1, n = 1 is proved at the beginning of the proof. Since m n is still a large number (finite), say l, it is sufficient as well as necessary to prove the result for a finite number l ( in the case considered, the binary variables are imbedded inside a tensor. Also, the polynomial representing the Boolean function is expressed through inner product operation over appropriate tensors ). Now, as an induction hypothesis, assume that the claim is true for l P = G2n B

(3.54)

variables. Since, every polynomial of (l + 1) variables can be written as a combination of two polynomials each of l variables, we have

Pf (X1 , X2 ,..., Xl + 1 ) = Pf 1 (X1 , X2 ,..., Xl ) + Xl + 1 Pf 2 (X1 , X2 ,..., Xl )

(3.55)

There are two possibilities, either Xl + 1 = + 1 or Xl + 1 = − 1. Hence, by the induction hypothesis (3.55), the system of linear equations in (l + 1) variables, becomes G2n G2n  P = G – G   2n 2n  From the recursive definition of Hadamard matrices,

we have

G2n G2n  1 1 G 1 = [ 1 ] ; G 2 = 1 – 1 ; G2n +1 = G – G    2n   2n P = G 2n + 1 B

(3.56)

(3.57)

Hadamard matrices are non-singular; thus, for any given f, a unique Pf exists (defined by a vector of coefficients).

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

49

In the language of tensor algebra, the same argument holds true except that the tensor can have ( the tensor utilized to couple the coefficients of the polynomial representing a Boolean function to the values of the polynomial) ‘0’ (zero) entries in addition to +1, –1 entries ( when contraction is performed over the appropriate indices). Uniqueness of such a polynomial is ensured by the uniqueness as a representation of Boolean function ( from the discussion/proof above ). Thus, in the tensor algebra notation, we have P = G ⊗B

(3.58)

where ⊗ denotes the inner product of two tensors.

Q. E. D.

It should be clear that the above representation theory has relevance to the minimum sum of products representation of a Boolean function. The above theory, as is easily seen holds true, if one is interested in finding the equivalent polynomial of a Boolean function which assumes {0,1} values. One way to see the result is by the following claim. CLAIM: Every monomial over {1, –1} can be written as a polynomial over {0,1} by the change of variable (BrB), x = 1 - 2 u, as follows: k

k

i =1

i =1

∏ Xi = 1+ ∑ (−2)i

∑ ∏U Si j ∈Si

j

(3.59)

with Si , a subset of {1, 2, ..., k } with i elements. For example, X1 X 2 = (1 − 2U1 ) (1 − 2U 2 ) = 1 − 2 (U1 + U 2 ) + 4U1U 2

The variables can be the components of a tensor X = (Xi 1,..., ik ), The representation theory developed above is now utilized for representing the multidimensional error correcting codes in a way that generalizes the representation described in section 3. Consider the linear multidimensional (m, k ; m, n) code C. The code can be represented by viewing each component of the codeword tensor as a Boolean function of utmost m k variables. A tensor V ∈ C , if and only if there exists an m-dimensional tensor of order k ( binary entries ) such that, with {1, – 1} tensor, X = (Xi 1,..., ik ),

Vi 1, i 2,..., in = fi 1, i 2,..., in (X ) .

(3.60)

The Boolean functions associated with the components of a linear multidimensional codeword tensor are determined by the generator tensor entries through which the code is represented. For linear multidimensional codes, every component of the codeword tensor fi 1,i 2,..., in (X ) correspond to an XOR operation of some variables of the information tensor ( determined by the corresponding entries of the generator tensor ). Thus, for every component (i1, i2,..., in), the Boolean function fi 1,i 2,..., in (X ) can be transformed by the method

50

Multidimensional Neural Networks: Unified Theory

described in Theorem (3.9) to an equivalent polynomial over {1, – 1} one monomial only.

mk

which consists of

Now, by the same argument as in Theorem (3.4), the maximum likelihood decoding (MLD) of a given received tensor word is equivalent to solving the following maximization problem:

Max (Wi 1,..., in ⊗ fi 1,...., in (X )).

(3.61)

By the procedure/reasoning through which we arrived at the above conclusion, tells us that the MLD problem as defined above also holds ( (3.61) holds) for non-linear multidimensional codes. For non-linear multidimensional codes, a component of the codeword tensor fi 1,i 2,..., in (X ) can consist of more than one monomial. Other than that, each component satisfies all the conditions to arrive at the above conclusion. From the above generalization, it follows that, for both linear as well as non-linear multidimensional codes, the maximum likelihood decoding problem is equivalent to the maximization of a multi-variate polynomial defined over the components of {1, –1} tensor i.e. over a tensor X with entries, of dimension m and order k . Hence, the following interesting theorem follows: Theorem 3.10: The following three problems are equivalent: (1) Maximization of multivariate polynomials with rational coefficients over the multidimensional hypercube, (2) Maximum likelihood decoding (MLD) problem of an (m, k ; m, n) multidimensional linear code, (3) Maximum likelihood decoding (MLD) problem of a not necessarily linear (possibly non-linear) multidimensional code, each of whose codewords are tensors of dimension m and order k . In view of the results in section 5 for non-binary codes which parallel those in section 3 for binary codes, maximum likelihood decoding of non-binary, nonlinear multidimensional codes is again equivalent to maximization of multivariate (variables being the components of a tensor) polynomial over a subset of multidimensional lattice. Various results (theorems, concepts, designs etc.) on optimization of multi-variate polynomials over various subsets of lattice were developed in various scientific fields such as electrical engineering, mathematics, computer science, operations research etc. These results are being translated to multidimensions and also the repercussions which follow immediately from the tensor linear operator are being documented. For instance, in one dimensional logic theory, various theorems including the representation of a Boolean function in the minimum sum of products (MSOP) form are well studied. In view of the results in (Rama 3), utilizing the fact that matrix linear operator is a special case of tensor linear operator, various theorems on

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51

multivariate polynomials ( imbedded in matrices ) are translated to the case where the polynomials/monomials are imbedded inside tensors. In as much as the linear space structure is utilized in deriving the results/theorems, the translation of the results from one dimensional logic theory to multidimensions is done with the generic principles described in (Rama 3), (Chapter 4). The results in sections 3, 4, 5, 6 effectively demonstrated the relationship between multidimensional codes, the energy functions optimized by multidimensional neural networks over various subsets of the lattice, optimization of multivariate polynomials (the terms/monomials of which are based on the generator, other tensors) over the various subsets of the multidimensional lattice. Thus, these local optima of the multivariate polynomials have the structure parallel to various linear transformation groups and basis of a certain linear spaces. Utilizing a natural leap of imagination, the author considers univariate as well as multivariate polynomials, power series in tensor variables with tensor coefficients. Specifically, an interesting problem that arises in structured Markov random fields is the problem of determination of tensor zeroes of the following univariate tensor polynomial, power series equations (Rama 6).

X 2 ⊗ A 2 + X ⊗ A1 + A0 = 0 m

∑X j =1

j

⊗ Aj =0

∞

∑ X j ⊗ Aj = 0

(3.62)

j =1

where X, {A} are tensors of compatible dimension, order such that the inner/outer product operations are well defined. The solution techniques developed in (Rama 11) when the linear operators are matrices are extended to the tensor linear operator case in (Rama 6). Also, various results that are well documented in the books such as (Gol) for matrix polynomials based on the properties of matrix linear operator are extended to tensor linear operator. Furthermore, in one dimensional system theory, various results are developed for systems of matrix polynomial equations utilizing only linear operator properties of a matrix. These results are extended to systems of tensor polynomial equations (Rama 3). In (Rama 6), the author formulates as well as solves the problem of determination of tensor variate zeroes of multi-tensor variate polynomial, power series equations L

L

i1= 0

m= 0

∞

∞

∑ ...∑ X ∑ ...∑ X

i1= 0

m= 0

i1 1

i1

1

⊗ X i22 ... X mim ⊗ Ai 1,...., in = 0 ⊗ X i22 ⊗ ... X mim ⊗ Ai 1,...., in = 0

(3.63)

Multidimensional Neural Networks: Unified Theory

52

Various other associated results are documented in (Rama 6). It is well known that the zeroes of a uni-variate scalar polynomial constitute a group. By utilizing the set of zeroes of a determinental polynomial associated with the uni-variate/multi-variate (tensor variables) polynomial, the set of tensor zeroes are divided into certain set of equivalence classes. Thus, a group structure is imbedded onto the linear subspace of tensor zeroes of uni-variate/multi-variate polynomial equations. Unlike the multivariate polynomials (whose terms/monomials are based on the components of tensors) optimized in sections 3, 4, 5, 6; in view of the above results, a natural question that arises is whether the local optimum of multi-tensor variate polynomials over various subsets of multidimensional (very high dimensional) lattice lead to (each variable is a tensor) codeword sets with better properties. When the information tensor, generator tensor, codeword tensors are blocked into sub-tensors and the objective function for the optimization problem over a subset of multidimensional lattice is rewritten, it is evident that a multi-tensor variate polynomial appears. Thus, such polynomials are subsumed in the ones considered in sections 3, 4, 5, 6. Integer Programming Problems: Solutions Using Decoding Techniques In computer science, operations research and other fields, problems of the following form arise very often:  n Maximize  ∑ Wi  i =1

∏X j ∈Si

j

  

(3.64)

where Sj is a subset of {1,2,...,n} and X j ∈{0,1} . Thus, the problem is concerned with optimizing a multivariate polynomial, whose variables assume integer values. By the discussion, in this section, every polynomial over {1, –1} can be transformed to an equivalent one over {0,1} by a change of variable. It is shown in section 2, that a special case of the above problem i.e. maximization of a quadratic form in {1, –1} variables arises in connection with the determination of global optimum stable state of a neural network and is equivalent to the minimum cut problem. This problem is known to be an NP hard problem. The problem in (3.64) was studied extensively by various researchers and the main effort concentrated in identifying the special cases which are solvable in polynomial time and in devising approximation techniques. The most common technique for solving the unconstrained {0, 1} program of the form in (3.64) is by transforming them to the problem of finding the maximum weight independent set in a graph, which is an NP-hard problem. The problem in (3.64) is transformed to the problem of finding the maximum weight independent set by using the concept of a conflict graph of a 0-1 polynomial. In (BrB), it is shown how decoding techniques can be utilized to maximize 0-1 nonlinear programs. The multidimensional version of the 0-1 nonlinear programming problem in (3.64) is given by

Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks

 n  Maximize  ∑ Wi ⊗ X ii   i =1 

53

(3.65)

where W, X are tensors containing the known coefficients w ‘s in W and the monomials in the variable components of the unknown tensor X. The inner product between these two tensors provides the scalar objective function whose variables are allowed to assume only {0, 1} or more generally finitely many values. It is shown in (Rama 6) that such an integer programming problem can be solved utilizing the multidimensional decoding techniques for linear block multidimensional codes. These results in operations research are avoided here and relegated to (Rama 6).

3.7 CONSTRAINED STATIC OPTIMIZATION In one/two independent dimensions, various static optimization problems are solved under the sub-fields of optimization theory such as (a) linear programming, (b) non-linear programming, (c) calculus of variations, (d) combinatorial optimization etc. With the innovative idea of formulating and solving the parallel problems in multidimensions (Rama 3) through the utilization of tensor linear operator (motivated by practical applications), vast literature in multidimensional optimization theory is generated. Various consequences of this innovative idea of the author are fully explained in the companion research article (Rama 3) on dynamic optimization. In the following, some innovative ideas of generic consequence in static optimization are described. In view of the results in section 5, the constraint set over which a multivariate polynomial (terms of the polynomial expressed in terms of the components of a generator tensor, received tensor in the case of MLD) is optimized is a subset of the multidimensional lattice (or bounded lattice, say, in multidimensions) and subsumes the multidimensional hypercube as its subset. These results naturally lead to a question as to whether it is possible to utilize the results in sections 3, 4, 5, 6 for optimizing multivariate polynomials over more general constraint sets in multidimensions. In the following theorems, constrained optimization over more general constraint sets utilizing the results of sections 3, 4, 5, 6 is discussed. Theorem 3.11: Consider a compact set in a multidimensional metric space. The local optimum of a multivariate polynomial (with the terms/monomials expressed in terms of the components of tensors and assuming binary/finitely many integer values) whose variables are allowed to assume finitely many values, over the compact set, occurs at the union of codewords of finitely many multidimensional non-binary/binary codes. Proof: From real/complex analysis (also topology), we have the Heine-Borel Theorem, which states that every open covering of the compact set ( in the space described by multiple independent variables ) has a finite sub-covering. The covering generally consists of open

54

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balls ( hyperspheres in multidimensional space) in every metric space ( although other sets could be utilized for covering ). But, it can be chosen to be a collection of convex hulls of bounded lattices in multidimensions. This (possibly countable) collection covers the compact set and thus has a finite sub covering. This implies that the constraint set chosen for optimization can be covered by finitely many bounded lattices (convex hulls of bounded lattices in multiple independent dimensions ). But, by the results of sections 3, 4, 5, 6 the local optimum of multivariate polynomials (terms/monomials expressed in terms of the components of tensors) over multidimensional bounded lattice (subsets of multidimensional lattice) constitute a linear/non-linear multidimensional codewords. Hence, the local optimum is achieved at the set of codewords of finitely many linear/non-linear codewords (tensors). Q. E. D. It should be noted that the determination of global/local optimum of the multivariate polynomial over the compact set is reduced to determining the global/local optimum of the energy functions of finitely many neural/generalized neural networks. It should be understood that some codewords may not be in the feasible region i.e. strictly inside the compact set. Also, when specific compact sets are chosen, further detailed information can be obtained on the local optima. Theorem 3.12: Optimization of a multivariate polynomial (with the terms/monomials expressed in terms of the components of tensors) over an arbitrary open set in a multidimensional space (metric space) is equivalent to the optimization over the union of codewords of countably many multidimensional linear codes or an infinite dimensional code. Proof: Let us consider an arbitrary open set in a multidimensional space (metric space). By the Lindeloff’s covering lemma, the open set can always be covered by utmost a countable collection of open balls or other sets. It is evident that the covering can be chosen to be a countable collection of convex hulls of bounded lattices (in multidimensional space). But, by the results in section 4, the local optima of the multivariate polynomial (the monomials/ terms being expressed in terms of the components of tensors) over a bounded lattice constitute multidimensional codeword set. Thus, the local optima of a multivariate polynomial occurs at the union of codewords of countably many multidimensional codes. Q. E. D. Remark Suppose, the compact set/open set (in multidimensions) is covered by finitely/countably many hyperspheres (multidimensional) and a quadratic/higher degree form is optimized. By the spectral representation theorem, the local optima of quadratic/higher degree form occur at the eigentensors with the eigenvalues being the corresponding values. This corresponds to L 2 norm based optimization. The above theorems illustrate two essential ideas of generic utility in static optimization: (a) optimization over more general constraint sets, (b) decomposition principle.

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In the following, the decomposition principle is explained as it is of generic utility in static optimization. Decomposition Principle Consider an arbitrary constraint set over which an objective function is optimized in one or more independent dimension variables. The constraint set is decomposed into the union of finitely many special sets with interesting structure. Optimization of various objective functions over the special sets has various interesting features: (a) various results are well known, (b) the local optima have interesting structure, (c) it is thoroughly studied etc. Utilizing these features, optimization of any objective function over the original set is decomposed into simpler problems. The above two theorems are only illustrative. The discovery and application of the above decomposition principle to multidimensional constrained optimization problems naturally led the author to investigate various other innovative ideas in static optimization. (I) Approximation of Objective Function by Polynomials, Power Series ( other Special Classes of Functions) Polynomials and power series (uni/multi-variate) are very important classes of functions. The optimization results (unconstrained as well as constrained) associated with these functions enable one to derive the local optimum of some classes of functions over various constraint sets invoking standard theorems (from approximation theory). For instance, the following theorem enables deriving results on continuous objective functions utilizing polynomials: Theorem 3.6 is utilized in association with the following theorem. Theorem 3.13: Every continuous function over a compact set always attains its maximum/ minimum over the set. Every continuous function can be arbitrarily closely approximated by polynomials ( multi-variate/univariate). Also, invoking the standard theorems from approximation theory, various classes of functions are arbitrarily closely approximated by polynomials: uni-variate/multi-variate. Thus, when these functions are utilized as objective functions, results associated with polynomials (derived in sections 3-6 ) are invoked. (II) Discovery of new local/global optimization techniques This requires utilizing either new classes of functions or new constraint sets. The constraint set structure renders the local optima of some functions with interesting structure and also the properties satisfied by the objective functions enables discovering efficient techniques. NP-Hard Problems: In computer science, operations research and other applied/theoretical research fields, various NP-hard problems are well identified and studied. It is well known that one NPhard problem is as complex ( in the terminology of complexity theory in theoretical computer

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science) as any other NP-hard problem. Finding algorithms which are efficient (in terms of complexity) for an NP-hard problem is well recognized as a difficult problem. The following is a difficult open problem in theoretical computer science: Problem: Does a polynomial time algorithm exist for an NP-hard problem? In other words, is the class of problems in NP, the same as the class of problems in P? i.e. is P = NP? In the following, an innovative algorithm/approach to solve various NP-hard problems in one dimension is described. The multidimensional generalization of this algorithm/ approach to any NP-hard problem (in multidimensions) is being formalized. It is an extension of the following results to multidimensions. In section 2, the problem of computation of minimum cut in a graph is shown to be equivalent to the problem of determining the global optimum of the energy function of a neural network i.e. maximizing a quadratic form over the hypercube. It is well known that this is an NP-hard problem. In the following, an attack on this problem is described. Positive Definite Synaptic Weight Matrix: Determination of Global Optimum Stable State of a Neural Network: Consider a neural network whose synaptic weight matrix is symmetric as well as positive definite. In the following, an algorithm to determine the global optimum stable state of such a neural network is described. (a) Utilizing the well known theorem in linear algebra, every positive definite symmetric matrix, S can be decomposed into the following form by means of Cholesky Decomposition. S = N NT (3.66) where N is a lower triangular matrix. (b) The quadratic form being optimized by the neural network over the hypercube can be expressed into the following form: (3.67) X T S X = XT N N TX = YT Y , where Y = N TX . Since S is positive definite, XT S X > 0. Thus, YT Y > 0. The scalar expression for the quadratic form n

in terms of Y is given by

∑Y j =1

2

j

. Thus, it is evident that the value of the quadratic form is either

maximized/minimized when the value of each term i.e. Yj = ∑ N kj X k is either maximized or k

minimized. But, since, N in the Cholesky decomposition is a lower/upper triangular matrix, each of the terms is a linear form. Thus, to maximize the quadratic form, each linear form maximum/minimum value (whichever is larger) is determined over the constraint set, hypercube. Thus, the original NP-hard problem (of maximization of a quadratic form over the hypercube) is reduced to several linear programming problems i.e. optimization of several linear forms over the hypercube. In this novel algorithm/approach for various classes of NP-hard problems (minimum

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cut computation in an undirected graph, knapsack problem etc.), the complexity of the algorithm is determined by (a) Complexity of determination of Cholesky decomposition of a positive definite symmetric matrix. Since there are various polynomial time procedures for the spectral decomposition, computationally well studied efficient algorithms are available, (b) Solving the linear programming problems related to optimization of linear forms ( maximization or minimization whichever leads to a larger value for the term) over the hypercube. It is well known that there are polynomial time algorithms for linear programming problems. In some problems that arise in operations research, communication theory etc., constraint set is a convex polygon/polytope (convex hull of various finite structures leading to convex sets bounded by hyperplanes) etc. and a quadratic/higher degree form is optimized over the constraint set. Then, by means of Spectral/Cholesky type decomposition of the positive definite symmetric linear operator (in one as well as multidimensions), various linear programming problems are solved through efficient polynomial time procedures. The computation of complexity of such procedures, efficient algorithms for NP-hard problems in one and multi-dimension, are being documented. When the connection matrix has other special structure efficient algorithms are found.

Linear Programming Problems: Decomposition Principle In the general framework of a linear programming problem, the constraint set is a convex polytope. By means of the decomposition principle, utilizing the hyperplanes bounding the feasible set, the convex polytope is expressed as the union of finitely many rotated, translated hypercubes. The linear objective function is converted into a quadratic objective function (reversing the technique utilized in the above algorithm) and the results from neural network theory are invoked to determine the local optima of the objective function (union of stable states of various neural networks). Thus, unlike the simplex algorithm, only a subset of the vertices of the feasible region that constitute the stable states of neural networks is searched for determining the global optimum for the linear program. In the following, an alternative algorithm for any NP-hard problem is formally described.

Hop-Skip Algorithm: Maximum Likelihood Decoding – NP-hard Problems This algorithm is designed/analyzed by the author for maximum likelihood decoding of linear block codes (Rama 1). From (BrB), and Theorem (3.3), maximum likelihood decoding of a received word , Y with respect to a graph-theoretic code is equivalent to

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finding the maximum of the energy function E of a neural network defined by the graph G (the weights on the edges of G are given by W = (–1) yi with all its threshold values equal to zero. But, it is well known that the local optimum of a quadratic form over the hypersphere occurs at the eigenvectors (eigentensors of the symmetric second order tensor) of the symmetric matrix (associated with the symmetric matrix) with the value of the quadratic form being the eigenvalue. Thus, maximum eigenvector of the symmetric matrix maximizes the quadratic form over the hypersphere. Thus, the sign structure (sign of the components of the vector) of the maximum eigenvector is utilized as the initial condition to run a neural network i.e. Mathematically, let X 0 be the vector given by X 0 = Sign ( X max ), where X max is the normalized maximum eigenvector and X 0 is the initial state in which the neural network starts. A is the symmetric matrix. The analysis of hop-and-skip algorithm is provided below. X T A X = (X – X0 + X0)T A(X – X0 + X0)

(3.68)

= (X – X0)T A( X – Xmax) + X0 A X0 + 2 XT0 A( X – X0 ) = λmax + (X – X0)T A(X – X0) + 2 X0T A( X – X0) = λmax + (X – X0)T A(X – X0) + 2 λmax X0T(X – X0) = n –λmax + (X – X0) T A(X – X0) + 2 λmax X0T X

(3.69)

The above manipulations enable one to compare the value of the quadratic form on the hypercube at any discrete time instant against the maximum value on the unit hypersphere. The particular choice of initial condition, minimizes the Hamming distance between the maximum eigenvector and the initial condition vector to run the neural network. The set of eigenvectors of the connection matrix of neural network span the entire space or a subspace of it. Similarly, the set of stable states/ stable vectors span the space or a sub-space. To determine the maximum stable state, the essential idea of the above approach is to find the vector closest to the maximum stable state and utilize it as the initial condition to run the neural network. Detailed analysis of the algorithm is being investigated. Dynamic Optimization In (Rama 3), certain multidimensional system, in discrete/ continuous time is described by the following state space representation through tensors: Discrete Time:

X(n + 1) = A(n) ⊗ X(n) + B(n) ⊗ U(n), Y(n) = C(n) ⊗ X(n) + D(n) ⊗ U( n).

(3.70)

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Continuous Time:

X(t + 1) = A(t) ⊗ X(t) + B(t) ⊗ U(t), Y(t) = C(t) ⊗ X( t) + D(t) ⊗ U(t).

(3.71)

where ⊗ denotes the inner product between compatible tensors in the system description in continuous/discrete time. Utilizing this state space representation, the author formalized a unified theory of control, communication and computation in multi/infinite dimensional systems, first discovered in (Rama1) for one dimensional systems. This theory enabled the author to develop a highly advanced version of the theory of evolution of life from organic matter. In this theory the author reasons that various body organs, functions of living systems have evolved over time and that bilogical systems are organic/inorganic matter based dynamical systems.

3.8 CONCLUSIONS Tensor linear spaces over finite fields are utilized to describe and study the structure/ properties of multi/infinite dimensional linear codes. The three concepts: multidimensional neural/generalized neural networks, multidimensional codes, multivariate polynomial (terms/monomials being expressed in terms of the components of generator, other tensors) optimization over various subsets of lattice, are related. It is shown that (a) the problem of maximum likelihood decoding of error correcting codes (multidimensional), (b) finding the global maximum of the energy function of neural/ generalized neural networks, and (c) solving integer/non-linear programming problems in multidimensions are related. The equivalence is proved for binary as well as non-binary cases. This equivalence naturally suggests utilizing the solvable cases of one problem to the equivalent problem and vice versa. Full capitalization of equivalence leads to various new results (Rama 6). The programming problem of multidimensional neural networks is solved. Several new heuristic procedures for NP-hard problems in multidimensions are suggested from the equivalence. The decoding techniques of various (multidimensional extensions of one dimensional codes) codes are utilized to find approximate solutions of NP-hard problems. Various innovative results in static optimization are described. Infinite dimensional generalization of the results is briefly described.

REFERENCES (Ara) B. Arazi, “Common Sense Approach to the Theory of Error Correcting Codes, “ MIT Press book. (BoT) A.I. Borisenko and I.E. Tarapov, “Vector and Tensor Analysis with Applications, “ Dover Publications Inc., New York, 1968.

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(BrB) J. Bruck and M. Blaum, “Neural Networks, Error Correcting Codes and Polynomials Over the Binary Hypercube, “ IEEE Transactions on Information Theory, Vol. 35, No. 5, September 1989. (Gaal) Gaal, “ Group Theory, “ Academic Press, 1982, (Gol) I. Goldberg, “Matrix Polynomials, “ Academic Press, 1972. (Rama 1) Garimella Rama Murthy, “Unified Theory of Control, Communication and Computation—Part-1, “ Manuscript to be submitted to the IEEE Proceedings. (Rama 2) Garimella Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory, Logic Synthesis, “ Published in International Journal of Neural Systems, Vol. 15, No. 3, pp 223-235, 2005. (Rama 3) G. Rama Murthy, “Optimal Control, Codeword, Logic Function Tensors: Multidimensional Neural Networks,” International Journal of Systemics, Cybernetics and Informatics, October 2006, pages 9-17. See also Chapter 4. (Rama 4) Garimella Rama Murthy, “Multi/Infinite Dimensional Logic Synthesis, “ Manuscript to be submitted to the IEEE Transactions on Computers. (Rama 5) Garimella Rama Murthy, “Signal Design for Magnetic and Optical Recording Channels, “ Bellcore Technical Memorandum, TM-NWT-018026. (Rama 6) Garimella Rama Murthy, “Tensor Variate Polynomials/Power Series, Tensor based Functions, Tensor Algebraic Geometry: Optimization, “ Manuscript to be submitted to the Transactions of American Mathematical Society. (Rama 10) Garimella Rama Murthy, “Unified Theory of Control, Communication and Computation: Dynamical Systems, “ Manuscript in Preparation. (Rama 11) Garimella Rama Murthy, “Transient and Equilibrium Analysis of Computer Networks: Finite Memory and Matrix Geometric Recursions, “ Ph. D. Thesis, Purdue University, West Lafayette, Indiana. (Rama 12) Garimella Rama Murthy, “Origin of Universe: Living/Non-Living: Grand-unification Theory of Universe, “ Manuscript in preparation.

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CHAPTER

4

Tensor State Space Representation: Multidimensional Systems

4.1 INTRODUCTION With the efforts of researchers in electrical engineering, linear system theory started with abstract models of arbitrary linear systems through forced/unforced nth order difference equations in discrete time and differential equations in continuous time. Such representations are called the input-output representations of the linear system. These arbitrary system (electrical, mechanical, chemical, hybrid systems) evolution equations were then converted into first order differential/difference equations in state, control, input, output vectors through state, input, output coupling matrices. Such a representation is called the state space representation. The state space equations take the following form (Gop) Discrete Time Systems:

X(n + 1) = A( n) X(n) + B(n) U(n), Y(n) = C(n) X(n) + D(n) U(n), Continous Time Systems : X(t) = A(t ) X( t) + B( t) U( t)

Y(t ) = C(t ) X (t ) + D(t ) U (t )

(4.1)

where {A(n), B(n), C(n), D(n)} as well as {A(t), B(t), C(t), D(t)} are matrices of compatible dimensions. Thus, in the design, analysis and synthesis of linear systems, linear algebra techniques were extensively utilized. Various, input-output representation related concepts such as impulse response, systems function were shown to be derivable from the state space description. Also new concepts such as controllability and observability are studied in terms of state space representation. Thus, the state space representation of linear systems proved to be a far better description of arbitrary systems.

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Motivated by various practical applications, multidimensional systems were then studied. Various system theorists tried to extend one dimensional system theory to multidimensions utilizing the ideas of local state and local control. For instance, consider a typical discrete time, two dimensional system. The evolution of a prototype linear model is described by the state updating equation

X(h + 1, k + 1) = A1X(h, k + 1) + A2 X(h + 1, k) + B1U(h, k + 1) + B2U(h + 1, k) where

X( h, k )∈Rn and U( h, k )∈Rm are the local state and local input value at (h,k) and

A1,A

2

, B1 ,B

2

are real matrices of suitable dimensions. This type of approach based

on local state and local control was utilized in association with partial differential equation based continuous time linear multidimensional systems. These representations of continuous time as well as discrete time multidimensional systems required considerable amount of ingenuity, careful tracking of the indices, in designing and analyzing such systems. To a certain degree, this notation impeded further progress in multidimensional system theory. With this type of approach/notation, modeling, design and analysis of certain linear/nonlinear, multi/infinite dimensional systems was a complicated task. The author for the first time realized that, for the evolution of CERTAIN multidimensional linear systems, tensor linear operator based state space description is necessary as well as helpful. This mathematically formal tensor state space representation was an important contribution for further progress in multi/infinitedimensional system theory (linear/non-linear dynamical systems). Also, the author after carefully observing various multi/infinite dimensional systems (explicitly stated as a static or dynamical system or when a proper abstraction is made the multidimensional nature of problem/ phenomenon becomes apparent) such as those that arise in multi/infinite dimensional neural networks (Rama 2), databases ( utilizing multiple attribute tree etc. ), multi/infinite dimensional coding theory (Rama 3), proposed the utilization of tensors ( of order, dimension finite/infinite ) as the linear operators in the design, analysis and synthesis of such systems. This idea is already utilized in some applications. It should be noted that in the analysis of some systems defined over finite fields and other discrete structures, utilization of tensors considerably simplifies the analysis. In the case of multidimensional systems, there is no natural notion of causality. Various types of causality ( quarter-plane causality, half-plane causality) are artificially imposed by different choices of neighbourhood sets. With such an approach (for all multidimensional systems), it is very difficult to study controllability, observability and stability. The author realized that for certain multidimensional systems, utilization of tensor linear operators to represent the state, control, input, output variables, is very convenient (from the point of view of design and analysis of such systems) (Rama 1).

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This chapter is organized as follows. In section 2, the conventional approaches to two/ multidimensional system theory are summarized. It is also described how the utilization of tensor linear operator associated with multidimensional linear spaces provides a new approach for formulating as well as solving the problems related to static as well as dynamical systems (defined over multidimensional linear space). In section 3, state space representation of certain multi/infinite dimensional linear systems utilizing the tensor linear operator is formally described. In section 4, it is illustrated how the utilization of tensor based state space representation enables one to translate the results from one dimensional systems to certain multidimensional systems. Various generic principles of how to translate the results from one dimensional system theory to multi/infinite dimensional system theory are provided. In section 5, multi/infinite dimensional time series analysis models are described. In section 6, utilizing the concepts of local state, local input, local control in the multi/infinite dimensional state space, various state space representations for multi/infinite dimensional distributed systems are formally described. These state space representations enable one to translate the results developed for conventional multi/infinite dimensional systems to those described through the tensor state space representation. The chapter concludes with some conclusions.

4.2 STATE OF THE ART IN MULTI/INFINITE DIMENSIONAL STATIC/DYNAMIC SYSTEM THEORY: REPRESENTATION BY TENSOR LINEAR OPERATOR One of the main tools in the design and analysis of one dimensional linear dynamic systems as well as static systems is linear algebra. Motivated by practical applications in image processing and other fields, system theorists proposed various input-output models for two/multidimensional systems. Models which exhibit quarter plane causality have been initially investigated from the input-output point of view (BiF) in the framework of two dimensional filter theory, where two dimensional filters are represented by proper rational functions in two indeterminates of the following type:

W (Z1 , Z2 ) =

∑nZ

i + j ≥1

1+

ij

1

i

Z2 j

∑dZ

i + j≥1

ij

1

i

Z2 j

(4.2)

The idea of associating two dimensional state space models with two dimensional filters was originated very naturally. However, since the beginning it appeared that the canonical technique based on the Nerode equivalence leads to an infinite dimensional state space. The reason was to utilize a matrix as the linear operator to describe the state dynamics. So, following some heuristic procedures, several finite dimensional models have been (BiF) introduced, where two notions of state play different roles: 1. local states: X(h,k) belong to a finite dimensional vector space. They enter in the state updating equation and determine the value of the output.

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2. global states Xn = {X (i + h , – i ) , i ∈ z} provide the initial conditions on a separation set of Z × Z. These belong to an infinite dimensional vector space (in one independent dimension), which provides an extension of the space of Nerode equivalence classes. The most common state space model with quarter plane causality is represented by the following equation.

X(h + 1, k + 1) = A 1X (h, k +1) + A2 X(h + 1, k ) + B1 u (h, k + 1) + B2 u (h + 1, k ) Y(h , k ) = CX(h , k )

(4.3)

where x(h , k )∈ R n , u( h, k )∈R m , y(h , k )∈R p , are the values of the local state, the input and the output at (h, k)∈Z × Z . Since the local state at (h+1, k +1) is computed by solving a first order difference equation, the system (4.3) denoted by Σ 1 = (A 1, A 2, B1, B2, C) is named a first order system. The above model has been extensively studied in its general form and under some conditions/constraints on the system matrices. The most popular particularized version of (4.3) is Roesser’s model, where the local state space X is assumed as the direct sum of two vector spaces Xh and Xv , and the matrices of the model are constrained to have the following form (partitioned) 0   A111 A121   0 B11   0  = = A1 =  B B , ,  , A2 =  2     2  1 2 (4.4) 2 0   0  A21 A22  0   B2  Second order models are less frequently used: the typical structure of their equation is given by

X ( h + 1, k + 1) = A1X ( h , k + 1) + A2 X ( h + 1, k ) + A0 X ( h , k ) + BU ( h , k ) Y (h , k ) = C X(h , k )

(4.5)

In Attasi’s model A 1 and A 2 are commutative matrices. Also, A 1A 2= –A 0. It realizes separable filters only and constitutes an interesting second order model, as the underlying theory is very close to the one dimensional theory (BiF). Recently, the behavior approach has been extended to two dimensional systems. Following this theory, a two dimensional system is defined by a family of β admissible functions (behavior), defined over the discrete plane. These functions are characterized by the property of belonging to the Kernel of a polynomial matrix M (Z1, Z2) in two variables

β = {ω = ∑ wij z1i z2j M ω = 0} i , j∈Z

(4.6)

Associated with the external description provided by the behavior different internal representations can be given by introducing the so called latent variable models. State variable models constitute a particular type of latent variables, that hold the memory of the system with respect to the notion of past introduced on Z × Z. When a state description is possible,

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i.e. when the notion of past, present and future are allowed by the structure of β , the behavior is called Markovian. Since there is not any natural direction for the evolution in Z × Z , the Markovian property appears more general than the familiar quarter plane causality and has been exploited in the analysis of non-causal two dimensional dynamics. Also, various static systems that involve simple linear transformations in the multidimensional space were previously abstracted utilizing the matrix linear operator. Such systems arise in practical applications such as databases (modeling storage of multiple attribute trees), computerized topography etc. The techniques developed for design and analysis of such systems were thus very elementary. The above efforts in two/multidimensional system theory were primarily utilizing the matrix linear operator on an n-dimensional ( in one independent variable) vector space. System theorists did not realize that utilization of tensor linear operator (in multidimensions) could lead to design and analysis of a large class of multidimensional systems. In the following areas, utilization of tensor linear operator to describe the multi/infinite dimensional state space enables one to formulate new problems , introduce new concepts, derive new results/theorems. Some of the areas of interest where such an idea could be utilized are (1) Multi/Infinite dimensional computation theory, (2) Multi/Infinite dimensional information/communication/coding theory, (3) Multi/Infinite dimensional rate distortion theory, (4) Multi/Infinite dimensional stochastic systems—Theory of Markov random fields, (5) Multi/Infinite dimensional time series analysis, (6) Multi/Infinite dimensional digital signal processing, (7) Theory of Multi/Infinite dimensional connectionist structures—graphoids, (8) Theory of databases utilizing multidimensional storage, (9) Matroid theory, (10) Multi/Infinite dimensional Game theory. By the utilization of the idea of capturing a multidimensional state space through a tensor linear operator, new research problems can be formulated and solved.

4.3 STATE SPACE REPRESENTATION OF CERTAIN MULTI/INFINITE DIMENSIONAL DYNAMICAL SYSTEMS: TENSOR LINEAR OPERATOR A multidimensional system transforms an m -dimensional tensor (array) of order r into a k dimensional tensor of order s. In the following, some confusion that arises in the terms utilized is cleared. Remark: Notation In the tensor notation, the word “dimension of a tensor” stands for the number of values each independent variable assumes, whereas the word, “order” represents the

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number of independent variables. In that sense, usage of the term “multidimensional” system seems incorrect compared to “multi-order” system. But, this is a matter of notation. To stick with familiar jargon, in the following the author utilizes the term “multidimensional systems”. From the context, the reader should be able to ascertain the usage of words, “order”, “dimension”. Infinite dimensional systems lead to further confusion. If each independent variable assumes infinitely many values ( in contrast to finitely many values assumed by each independent variable in a multidimensional system ) and there are only finitely many independent variables, the system description utilizes infinite dimensional tensors of finite order for state, input, output variables. But, if the number of independent variables is also infinite, the dimension as well as order of tensors utilized in the representation of variables is infinite. It should be noted that, in the case of discrete time systems, each component of the tensor input, output, state variables is a function of a discrete time index. But, in the case of continuous time systems, each component of the tensor input, output, state variables is a function of the continuous time index. Also, in the case of time varying systems, the transformation is a function of the index (discrete or continuous), whereas in the case of time-invariant systems, the transformation is independent of the index. Definition A dynamical system is linear if and only if, given any two points (scalar, vector, tensor variables) in the input space, say U1 and U2, and given any two scalar ( real or complex ) constants, the following property is satisfied by the transformation L, describing the dynamical system: L (C 1U 1 + C 2U 2 ) = C 1L (U 1 ) = C 1L (U 1 ) + C 2L (U 2 ); C 1 , C 2 ∈C or R or any field

(4.7)

If the above property is violated by the dynamical system, we call it a non-linear system. Conventionally, in multidimensional ( multi-order may be more appropriate, but is not utilized by the author ) system theory, in the case of discrete time dynamical system (an example is provided in section 2), the evolution is described by means of local state, local control, local input and local output variables. This is very cumbersome. In the case of certain multidimensional systems, the state space representation by means of tensors (described below) enables one to compactly capture a higher order difference equation through TENSOR notation. In order to describe the tensor state space representation, the following concepts/ideas from tensor analysis are explained.

Concepts from Tensor Analysis Tensor Function of a Scalar Argument It is a rule assigning a unique value of a tensor to each admissible value of a scalar t (BoT). The variable t can be a discrete index assuming countably many values or a

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continuous index assuming uncountably many values. To indicate such a function, we write Ai 1,i 2 ,...,in = Ai 1, i 2 ,..., in ( t ) (4.8) For instance, the state of stress of an elastic medium varies in time. Then, the stress tensor becomes a function of time i.e. Pik = Pik (t) (4.9) By the derivative of the function (4.8) with respect to time/index, we mean the tensor with the components,

d Ai 1, i 2,..., in (t) dt

=

Lt Ai 1, i 2,..., in (t + ∆t ) − Ai 1,..., in (t ) ∆t → 0 ∆t

(4.10)

calculated in a coordinate system which does not vary in time. The derivative is clearly of the same order as the tensor itself. With the above notation from tensor analysis, certain multi/infinite dimensional discrete time/index dynamical system can be described by means of a state space description of the following form:

Tensor State Space Representation of Certain Discrete Time Systems Discrete Time Systems:

X( i 1,..., ir ) (n + 1) = A( i 1,..., ir ; j1,..., jr ) (n) X( j 1,..., jr ) (n) + B( i 1,..., ir ; j1,..., jp ) (n)U( j 1,..., jp) (n), Y(l 1,...,ls ) (n) = C(l 1,..., ls ; j 1,..., jr (n) X( j 1,..., jr ) (n) + D(l 1,..., ls ; j 1,..., jp ) (n)U( j 1,..., jp ) (n).

(4.11)

where A(n) is an m dimensional tensor of order 2r (called the state coupling tensor ), X(n) is the state of the dynamical system at the discrete time index n, whereas X(n+1) is the state of the system at the discrete time index n+1. Furthermore B(n) is an m dimensional tensor of order r+p ( called the input coupling tensor ), Y(n) is an output tensor of dimension m and order s. U (n) is an m dimensional input tensor (varying with the discrete time index of order p) and C(n) (called the state coupling tensor to the output dynamics) is an m dimensional tensor of order (s + r), D(n) is the input coupling tensor to the output dynamics of dimension m and order s+p. In the above state space description of certain type of multidimensional discrete time dynamical system, there are r dimension variables which are inherently discrete. The evolution of the system (changes in the system parameters) occur at discrete time instants. The notation for index set in the state equations requires some explanation. Since the state tensor is an m -dimensional tensor of order r, it will have m components. When the system evolves, it transits through tensors in the state space. With the summary of tensor functions of scalar argument provided above, the dynamics of certain type of multi/infinite dimensional continuous time/index systems is described by the following state space description:

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Tensor State Space Representation of Certain Continuous Time Systems i X( i 1,..., ir ) (t) = A( i 1,..., ir ; j1,..., jr ) (t) X( j 1,..., jr ) (t) + B( i1,..., ir ; j 1,..., jp ) (t) U( j1,..., jp ) (t), Y(l 1,..., ls ) (t) = C(l 1,..., ls ; j 1,..., jr ) (t) X( j 1,..., jr ) (t ) + D(l 1,..., ls ; j 1,..., jp ) (t)U( j 1,..., jp ) (t ).

(4.12)

where A(t) is an m dimensional tensor of order 2r (called the state coupling tensor to the state dynamics), X (t) is the state of the dynamical system at the continuous time/ . index t, whereas X (t) is derivative of the state of the system. Furthermore, B (t) is the m dimensional tensor of order r+ p (called the input coupling tensor to the state dynamics), Y (t) is the output tensor of dimension m and order s. Also, U (t) is an m dimensional input tensor of dimension m and order p , and C (t) is an m -dimensional tensor of order (s + r), D (t) is the input coupling tensor to the output of dimension m and order s + p . It should be noted that the state space description provided above for certain continuous/discrete index systems hold true even for certain infinite dimensional systems. In the case of infinite dimensional systems, in the state space descriptions, the tensors utilized are of dimension/order infinity ( either or both of them). Now, the above tensor state space representations are contrasted with the conventional approaches in the representation of certain multidimensional systems. It is reasoned that the Tensor State Space Representation is an important leap in multi/ infinite dimensional system theory. Also, another objective is to remove the confusion in the mind of the reader who read the classical literature in multi/infinite dimensional system theory with matrix linear operator notation. The primary source of confusion is not so much in the discrete time/index multidimensional systems, but in the case of continuous time /index multidimensional systems. Conventional Multidimensional System State Space Representation versus Modern Tensor State Space Representation: In section 2 as well as section 3, the limitations of the way system theorists tried to represent and analyze the two/multidimensional discrete time/index systems is discussed. Also, the advantages of tensor state space representation (of certain large class of multi/ infinite dimensional systems) discovered and formalized by the author are described. The transition from the conventional mode of thinking where the system is represented by means of multiple independent variables, local state/local control are coupled to the system dynamics by means of matrices to the modern version where tensor notation is utilized, requires the realization that the linear space utilized in multidimensions is captured through the tensor and the system dynamics when done in discrete time requires a discrete variable. The continuous index case requires more imagination to understand the transition from conventional approaches to the modern approaches. In the conventional multidimensional system representation, partial differential equations are utilized to describe the input-output behavior as well as the state (internal description) dynamics. In the conventional approaches, multiple independent variables are tracked through separate indices, leading to partial differential equations. But, the utilization of tensor linear operator

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and the tensor function of scalar argument enables one to describe the dynamics of tensor state variable as a function of one continuous time/index variable. Thus, the discrete as well continuous multi/infinite dimensional system state space representation utilizing tensors resembles the familiar one dimensional system state space description. The above tensor state space description reduces to the one dimensional case when the order of the tensors is one. Thus, various results developed on one dimensional linear spaces for one dimensional linear systems are readily translated to certain multi/ infinitedimensional systems described through tensor linear spaces (with some care taken in pathological cases as well as when the problem being solved depends heavily on the neighborhood set).

4.4 MULTI/INFINITE DIMENSIONAL SYSTEM THEORY: LINEAR DYNAMICAL SYSTEMS – STATE SPACE REPRESENTATION BY TENSOR LINEAR OPERATORS The state space representation of one dimensional linear systems resembles that in (4.11), ( 4.12). In fact, one dimensional linear systems are a very special case of certain multi/ infinite dimensional systems described through (4.11), (4.12). A natural question that arises is whether it is possible to transfer the results from one dimensional systems to certain multidimensional systems described through (4.11), (4.12). It is explained in the following that it is possible to do such a translation provided some care is taken in deriving the results for certain class of multi/infinite dimensional systems. Some principles which can be utilized as a guideline in deriving the results for multi/infinite dimensional systems are provided below: (1) In the case of one dimensional systems utilizing the state space representation of a linear system, if a result is derived on the system response (invoking the standard theorems in the theory of ordinary difference/differential equations), that result has a corresponding version for multi/infinite dimensional systems when the inner product and outer product between state vector/input vector/ output vector, matrices appearing in the state space descriptions are replaced by those between compatible tensors in multi/infinite dimensions. One must exercise care in making sure that the tensor products make sense. (2) The tensor state space representation (rather than vectors and matrices in one dimensional case) enables one to translate the results on controllability, observability, stability from one dimensional linear space based dynamical systems to certain multidimensional linear space based dynamical systems. The tensor state space representation enables one to translate various problems for one dimensional systems, in a one to one manner to certain multi/infinite dimensional systems. These problems are defined utilizing the state space structure to be linear (linear spaces in one/multi/infinite dimensions). In translating the solution

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of the problem from one dimension to multidimensions, inner product/outer product between vector-matrix variables are replaced by those between tensortensor variables. Care should be taken to ensure that the problem statement in multidimensions doesn’t utilize the neighbourhood structure. (3) The multi/infinite dimensional state space structure is such that there is no notion of causality. From 1970s, system theorists, electrical engineers, computer scientists developed various notions such as quarter plane causality, half plane causality, other types of causality ( to introduce some form of ordering on the two/multidimensional state space ) for providing an input-output description. But the state space representation through tensors (of certain multidimensional systems) enables one to get the associated input-output description as a special case ( for such systems ). Thus, various problems in image processing, database theory, theory of random fields are reformulated utilizing the tensor state space description and solved in this context. When these problems have multi/infinite dimensional state space structure (implicitly or explicitly specified) imbedded into the statement, utilizing the tensor linear operator (or the theory of tensor linear spaces) and the results in this chapter, they are considered to be solved. The systems in which problems are formulated can be static or dynamic. It should be reminded that various problems in different scientific disciplines (as listed in section 2) which are based on multi/infinite dimensional description are effected by the tensor state space description for linear dynamical systems. Even static systems where the state space structure is a multidimensional linear space, utilization of tensor linear operator, tensor algebra techniques provide convenient tools for formulation as well as solution of them. The above generic principles are easily illustrated with the typical problem of response determination of certain multi/infinite dimensional linear systems (whose dynamics are captured through the Tensor State Space Representation ). Details are avoided for brevity.

4.5 STOCHASTIC DYNAMICAL SYSTEMS In the following, multi/infinite dimensional versions of time-series models are discussed. They are the multi/infinite dimensional versions of Auto-Regressive (AR), Auto-Regressive Moving Average (ARMA) models. The models are formally described utilizing the tensor linear operator for the variables. The discrete time, multi/infinite dimensional versions of AR, ARMA models are given by

Yi 1,..., ir (n + 1) = Ai 1,..., ir ; j 1,..., jr (n) ⊗ Yj 1,..., jr (n) + Wi 1,..., ir (n),

(4.13)

Yi 1,..., ir (n + 1) = Bi 1,..., ir ; j 1,..., jr ⊗ Yj 1,..., jr (n) + Vi 1,..., ir (n) + Ci 1,..., ir ⊗Vj 1,..., jr (n − 1) + Di 1,..., jr ⊗ Vj 1,..., jr (n − 2)

(4.14)

where ⊗ denotes the inner product and the variables such as⊗Yj 1,..., jr (n) are tensors. The

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noise models Wi 1,..., ir (n), Vi 1,...,ir (n) are multidimensional versions of white noise. As in one dimension, the continuous time versions of these models are based on utilizing a continuous time index t, in the place of discrete time index n and replacing the noise models in (4.13 and 4.14) by the continuous time white noise or colored noise models. The formal description is avoided for brevity. The above models (which effectively reduce to the one dimensional models in the one dimensional case) enable one to derive various important details related to such stochastic processes in multi/infinite dimensions. For instance, the autocorrelation tensors, the power spectrum are derived based on the well known techniques for one dimensional systems. It should be noted that the multi/infinite dimensional power spectrum estimation problem (formulated using local state etc.) was well known to be very difficult. Thus, the utilization of tensor linear operators in certain multidimensional systems enabled one to invoke the results from one dimensional systems to be extended to certain multidimensional systems. Various interesting identities arise in the actual analysis. The details are avoided. In the following, state space representations for arbitrary stochastic linear systems are described. In one dimension, it is well known that the widely utilized Markov chains constitute the one dimensional stochastic linear systems. Thus, there has been research effort to extend the idea, approach to multi/infinite dimensions. Like the deterministic multi/infinite dimensional linear systems, conventionally various models based on the local state approach were developed. These are traditionally called the random field models. With the Tensor State Space Representation (TSSR) (of certain multidimensional systems) provided in section 3, stochastic multi/infinite dimensional linear systems, called structured Markov random fields, are based on the tensor linear operator. In the spirit of the one dimensional approach, the multi/infinite dimensional structured Markov random fields are homogeneous stochastic linear systems, described by difference equation of the following form in the discrete time/index

∏

( n + 1)

= ∏ ( n ) ⊗ P( n )

(4.15)

where Π(n) is the tensor of probabilities of the states in the state space, P (n) is the state transition tensor of the discrete time structured Markov random field. When the structured Markov random field is homogeneous, then P(n) = P . Both P(n), P are stochastic tensors. In the continuous time, the multi/infinite dimensional structured Markov random field is described by means of a generator tensor. It is given by •

d

∏ (t ) = dt π (t) = π (t) ⊗ Q(t)

(4.16)

where Π(t) is the tensor of probabilities of states in the state space at time t, Q (t) is the generator tensor of the continuous time strucured Markov random field. Q(t) satisfies the properties of a generator tensor. The equilibrium distribution of states in the discrete as well as continuous time/index

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structured Markov random field are derived through the utilization of the spectral representation theorem of the linear operator (tensor) utilizing the eigenvalues and eigentensors of the linear operator. When the state transition tensor as well as generator tensor have the G/M/1-type structure, M/G/1-type structure (Neu), the invariant distribution of the random field has the tensor geometric form. The derivation of the form of invariant distribution and efficient recursions for the invariant distribution follow from a generalization of the results in one dimension. In the following, state space representations for various types of multidimensional stochastic dynamical systems that are commonly utilized in electrical engineering are discussed. In the discrete time, the multi/infinite dimensional dynamical system is described by the difference equation of the following form:

X(n + 1) = A(n) ⊗ X(n) + B(n) ⊗U(n) + W (n) Y(n) = C(n) ⊗ X(n) + V (n) + D(n) ⊗ U(n)

(4.17)

The tensors A(n), B(n), C(n), D(n) and the state, input, output tensors are of compatible dimension and order. The noise terms are multi/infinite dimensional extensions of the independent, identically distributed noise model in one dimension. It is based on the following tensor based random variable/random process (like vector random variables, vector random processes) specification. Generally, they are zero mean tensors (each component random variable has zero mean) and as a sequence constitute independent tensor random variables. This model is the simplest model that is commonly utilized in stochastic control theory (ZoP), (SaW). Utilizing Tensor State Space Representation (TSSR), Unified Theory of Control, Communication and Computation is formalized in (Rama 4).

Co var iance tensor {W (m), W (n)} = Q(m) δ (m − n), Co var iance tensor {V (m), V (n)} = R(m) δ (m − n), Co var iance tensor {W (m), V (n)} = 0

(4.18)

These plant noise and measurement noise models are assumed to be independent of the normal random initial state tensor, X( ). The continuous time multi/infinite dimensional stochastic models utilize continuous time I.I.D. noise (as in one dimension). The state space model description has an additive I.I.D. noise term to those described in section 3. With the above state model, theorems in one dimensional stochastic control are extended to multi/infinite dimensions, since the matrix linear operator is replaced by the tensor linear operator. In translating the results inner/outer product between vectors/matrices are replaced by those between the tensors/tensors. Now, we consider a noise model which describes processes which are more complicated than the ones considered previously. The colored noise model considered in ARMA time series model is a special case version of the following noise model. In this model, the noise processes constitute a structured Markov random field in multi/infinite dimensions. The

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plant noise model and measurement noise are uncorrelated/independent. The noise models satisfy the following equations.

X(n + 1) = A(n) ⊗ X(n) + B(n) ⊗U(n) + L(n) Y (n) = C(n) ⊗ X(n) + M(n) + D(n) ⊗ U(n)

(4.19)

L(n), M(n) are discrete time structured Markov random fields. The fact that Markov random field is a stochastic linear system enables one to apply the stochastic dynamic programming. In the above noise model, the plant and measurement noise are made to be the most general models that are conceivable, while at the same time they are tractable. The continuous time version of the state space model has an additive term added to those in section 3. With the above state space representation, various results developed in one dimensional stochastic control theory (SaW) are extended to multi/infinite dimensional systems utilizing the generic principles described in section 3. Thus, various recursive forms for state estimation, filtering and prediction are translated from one dimensional systems to multidimensional systems, particularly with the I.I.D. form of noise. The time series model discussed at the beginning of the section with tensor state space representation, led the author to provide very detailed linear prediction type results in multi/ infinite dimensions when the noise process is white as well as colored. Thus, the linear prediction theory, which was so successful in theoretical as well as practical applications is successfully (in mathematical completeness) advanced to multi/infinite dimensions by the author with the tensor state space representation. The mathematical equations look familiar with tensor products being utilized in the equations. It should be noted that using the signal and noise models described in this section, multidimensional versions of Wiener and Kalman filters can easily be derived. Various results on estimation, prediction and control are translated from one dimension to multidimension (Rama 4) (when the multidimensional system has Tensor State Space Representation i.e. TSSR). In summary various results developed in one dimensional stochastic control theory, theory of one dimensional random processes are extended to multi/infinite dimensions through the Tensor State Space Representation.

4.6 DISTRIBUTED DYNAMICAL SYSTEMS Distributed dynamical systems are a class of systems which are more general than the dynamical systems considered above in some sense. They arise in various practical applications such as the electrical transmission lines (distributed inductance, capacitance, resistance along the line), image models, models of tomographic images of brain etc. One/multi/infinite dimensional systems in which the tensors which appear in the system dynamics that vary with time are one of the simple illustrations of distributed dynamical systems. These systems illustrate a form of non-homogeneity in the evolution

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of the system in the state space i.e. a dependence on the discrete/continuous time index of the manner in which the state coupling, input coupling, output coupling tensors vary with time, resulting in a distributed nature of the manner of state transitions depending on the location i.e. discrete/continuous time index. This naturally motivates considering systems, based on practical applications, in which the state transitions in multi/infinite dimensions depend on the location. This is once again reminiscent of the conventional models of two/ multidimensional signal processing. To formally provide models of distributed dynamical systems in multi/infinite dimensions, the following notation from tensor algebra/analysis is introduced.

Tensor Functions of Multiple Arguments It is a rule assigning a value of a tensor B to each admissible value of a set of variables (t1, t2 ,..., ts ). To indicate such a function, we write

Ci 1,..., in = Ci 1,..., in (t1 ,..., ts )

(4.20)

In the models of distributed systems described in the following, utilizing tensor linear operators, the state, input, output variables are functions of multiple discrete time/index or continuous time/index. The following concept from tensor analysis is also extremely helpful. Tensor Field: By a tensor field, we mean a rule assigning a unique value of a tensor to each point of a certain volume V ( V may be all of space). Let r be the radius vector of a variable point of V with respect to the origin of some coordinate system. Then, a tensor field is indicated by writing

Ai 1,..., in = Ai 1,..., in (r )

(4.21)

if the tensor is of order n. A special class of tensor fields are nonstationary fields, which are functions of both space and time i.e. of both the vector r and the scalar t:

ϕ = ϕ (r , t ), A = A(r , t )

(4.22)

A tensor field is said to be homogeneous if it has no spatial dependence. In this case, the above reduces to A = A(t)

(4.23)

Tensor fields which are continuous are of utility in physical applications and in modeling various real life dynamical systems. Non-stationary fields are of utility in modeling distributed dynamical systems. It will be evident to an intelligent reader, how the above concepts are utilized in the following models of distributed dynamical systems. Particularly, tensor fields enable one to define dynamical systems over regions in the higher dimensional space which are not

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bounded by hyperplanes. Such systems are of importance in various practical applications.

Quarter Plane Causal Distributed Dynamical Systems Motivated by the quarter plane causal model familiar from conventional two dimensional system theory, the author defines the following model of a linear system distributed in the plane or two dimensional distributed system. It is given by (1)

(2)

Xi 1,..., ir ( h + 1, k + 1) = A i 1,..., ir ; j 1,..., jr ( h, k + 1) ⊗ X j 1,.., jr (h , k + 1) + A i 1,..., ir ; j 1,..., js ( h + 1, k ) ⊗ X j 1,..., js ( h + 1, k ) + B 1i 1,..., ir ; j 1,..., jt ( h + 1, k ) ⊗ U j 1,..., jt ( h + 1, k )

Yi 1,..., ip (h , k ) = Ci 1,..., ip ; j 1,..., jr ( h, k ) ⊗ X J 1,..., jr ( h, k )

(4.24) (4.25)

where X(h, k ), U(h,k), Y(h,k) are the values of the local state tensor, input tensor and output tensor at (h, k) ∈ Z × Z. The multidimensional extension of this model is described based on the same spirit in the sense that the nearest/farthest neighbourhood set is partitioned into causal/non-causal parts and utilizing it in writing the multidimensional difference equation describing the system dynamics. For instance, the half plane causal model familiar in two dimensional signal processing is written utilizing the tensor linear operator in the same spirit as the above quarter plane causal model. The spirit in which various notions of causality is introduced into the system evolution in the state space is by means of natural/artificially induced decomposition of the state space. The state space is partitioned into neighborhoods and the dynamical system is described by means of a difference equation (multi/infinite dimensional) of the following form

X{( i 1,..., in )( k )∈( N + 1)} = A(i 1,..., in ; j 1,..., jn )( k ) ⊗ X {( j 1,...., jn )(k )∈ N} + B( i 1,..., in ; j 1,..., jn )( k) ⊗ U{( j 1,..., jn )( k )∈N}

(4.26)

Y{(i 1,..., in )(k )∈N} = C( i 1,..., in ; j 1,..., jn) (k ) ⊗ X {( j 1,..., jn) (k )∈N } + D( i 1,..., in ; j 1,..., jn) (k ) ⊗ U{( j 1,..., jn )( k )∈N }

(4.27)

where N, N+1 are neighbourhood sets in the multi/infinite dimensional state space which are not necessarily bounded by hyperplanes (captured by a structure like tensors/matrices). The above state space description of a dynamical system in discrete index variables is in the most general format conceivable. The advantages of such a model is the ability to make an arbitrary choice of the neighbourhood. If the neighborhood is chosen to be one among those in the set utilized for embedding causality structure onto the state space, various models result. The continuous time version of the above model utilizes, non-stationary tensor fields. The typical system evolution equations are given by i X{( i 1,..., in )( t)∈( N + 1)} = A(i 1,..., in ; j 1,..., jn )(t ) ⊗ X {( j 1,..., jn )(t )∈ N } +

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B( i 1,..., in ; j 1,..., jn )(t) ⊗ U{( j 1,..., jn )(t)∈N}

(4.28)

Y{(i 1,..., in )(t )∈N } = C( i 1,..., in ; j 1,..., jn) (t ) ⊗ X {( j 1,..., jn) (t)∈N} + D( i 1,..., in ; j 1,..., jn) (t) ⊗ U{( j 1,..., jn )(t)∈ N}

(4.29)

i where, Xi 1,..., in (t) is the tensor of partial derivatives (like the Jacobian matrix, we can call it a Jacobian tensor) of Xi 1,..., in (t) . Once again this is the most general model conceivable. If the neighbourhood set is represented by a tensor, we have a very important special case. If one has understood carefully, the notions of local state, local control and the essential ideas of the theory of ordinary/partial difference/differential equations, many results developed in those fields have been adopted to the case where vector-matrix variables are replaced by the tensor-tensor variables. The outcome of this mathematically formal approach is: (i) Results developed by the differential/partial differential equations community are adopted to the tensor-tensor based equations. Once again, the translation is done with relative ease, (ii) Distributed dynamical systems are modeled by using the half plane, quarter plane causal type neighbourhood models. In these models, the matrices/vectors are replaced by tensors. Various other models based on local state, local control, various types of decompositions of the state space that arise in fields such as image processing, tomography etc. are translated to the multidimensional case by replacing the vectors/ matrices by tensors. Various types of problems formulated and solved in conventional two/ multidimensional system theory are adopted to the tensor based difference/differential equations by utilizing tensor products and tensor algebra/analysis. Some illustrations of design, analysis of distributed systems are reported utilizing the tensor linear operators for local state, local control, local input, local output variables and replacing the vector/ matrix products by means of tensor-tensor products. They are avoided here for brevity.

4.7 CONCLUSIONS Utilization of tensor linear operator associated with dynamic as well as static linear systems enables one to formulate as well as solve various known as well as new problems utilizing the powerful tools of tensor algebra (Rama1). This important representation invoked by the author is hoped to have useful effect on various scientific/mathematical fields. State space representation by tensor linear operators is discovered and formalized (Rama1). It is formally demonstrated how the theory of certain multidimensional systems is developed utilizing the tensor state space representation and translations of the results from one dimensional system theory. Approaches to translate one dimensional stochastic control theory to multi/infinite dimensional systems are briefly described. New state space representations for distributed dynamical systems are developed which enable translating the results from conventional state space models of multidimensional systems. Thus, in

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essence the tensor linear operator based representation of static as well as dynamic systems has important impact on various fields of scientific endeavour.

REFERENCES (BiF) M. Bisiacco and E. Fornasini, “Optimal Control of Two Dimensional Systems,” SIAM Journal of Control and Optimization, Vol. 28, pp. 582-601, May 1990. (BoT) A. I. Borisenko and I. E. Tarapov, “Vector and Tensor Analysis with Applications,” Dover Publications Inc., New York, 1968. (Gop) M. Gopal, “Modern Control System Theory“, John Wiley and Sons, New York. (Neu) M.F. Neuts, “Matrix Geometric Solutions in Stochastic Models”, Marcel-Dekker, Baltimore. (Rama 1) Garimella Rama Murthy, “Tensor State Space Representation: Multidimensional Systems, International Journal of Systemics, Cybernetics and Informatics (IJSCI), January 2007, page 16-23 (Rama 2) Garimella Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory,” International Journal of Neural Systems, Vol. 15, No. 3, June 2005. (Rama 3) Garimella Rama Murthy, “Multidimensional Neural Networks: Multidimensional Coding Theory:Constrained Static Optimization” Proceedings of 2002 IEEE International Workshop on Information Theory. (Rama 4) “Optimal Control, Codeword, Logic Function Tensors: Multidimensional Neural Networks, IJSCI, October 2006, Pages 9-17. (SaW) Sage and White, “Optimal Control Theory,” Academic Press. (Zop) R. Zoppoli and T. Parisini, “Learning Techniques and Neural Networks for the Solution of N-stage Non-linear No n-quadratic Optimal Control Problems,” Topics in 2-d System Theory, 1992.

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CHAPTER

5

Unified Theory of Control, Communication and Computation: Multidimensional Neural Networks

5.1 INTRODUCTION In the mid 1940s, Norbert Wiener coined the word Cybernetics for the research field dedicated to understand the control, communication, computation and other such functions of living systems. It is well agreed that these functions of living systems are controlled by various functional sub-assemblies in the brain synthesized through bio-chemical circuits. Research work on this field was pursued by several researchers in diverse fields. The multidisciplinary effort resulted in progressing the literature on the subject. But no formally precise discoveries were made. Also, starting in 1950s, the research efforts in electrical engineering discipline led to the isolated theories of control, communication and computation. The central goal of these three fields is summarized in the following: • The problem of communication is to convey a message from one point in space and time to another point in space and time as reliably as possible. • The problem of control is to move a system from one point in state space to another point in state space such that a certain objective function is minimized • The problem of computation is to process a set of input symbols and produce another set of output symbols based on some information processing operation. These three problems, on the surface seem to be unrelated to one another. Also, in the mid 1960s, several researchers became interested in the mathematical model of the nervous system. This effort was meant to complement the research in cybernetics. Hopfield/ Amari succeeded in providing an abstract model of associative memory. Based on this abstract model, researchers are led to the following question which remained unanswered. Question: Is it true that the functional units responsible for control, communication and computation are synthesized through a network of homogeneous neurons?

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Occasionally research efforts led to establishing some relationship between the three fields. But, in this chapter it is shown (with mathematical clarity and preciseness) that in the sense of optimization ( consolidating the earlier efforts of other authors) of some objective function, these three problems are related to one another leading to one form of unification. From a practical point of view, this unification leads to design of brain of powerful robots. With the efforts of the author, Boolean Logic theory was generalized to multi/infinite dimensions using an optimization approach (Rama 1). This approach led to the area of multidimensional neural networks (Rama 1). Also using the generalization of results in (BrB) in one dimension, multidimensional linear as well as non-linear codes are related to multidimensional neural networks. Thus using these results the research fields: Computation and Communication are related through the common thread of neural networks. In this paper, the main achievement of the author is to show that optimal control tensors of certain multidimensional systems are synthesized as the stable states of neural networks. Thus utilizing the results summarized in this paragraph, Unified Theory of Control, Communication and Computation is generalized to multidimensional systems. This chapter is organized in the following manner. In section 2, unification of control, communication and computation in one dimensional systems is summarized. In Section 3, the discovery and formalization of Tensor State Space Representation of certain multidimensional systems is briefly discussed. Using this representation, optimal control tensors (in a well known criteria of optimality) are shown to constitute the stable states of a multidimensional Hopfield neural network. In Section 4, utilizing the results in (Rama 1), (Rama 2), Unified Theory of Control, Communication and Computation in multidimensional systems is formally described. Conclusions are reported in Section 5.

5.2 ONE DIMENSIONAL LOGIC FUNCTIONS, CODEWORD VECTORS, OPTIMAL CONTROL VECTORS: ONE DIMENSIONAL NEURAL NETWORKS Researchers such as Hopfield realized that an associative memory is associated with optimizing a quadratic form over the hypercube. Other authors also realized that the concept of a logic gate (CAB) (in one dimension), concept of error correcting code (BrB) could be related to one dimensional neural networks (optimizing a quadratic/higher degree form). These efforts are summarized in the following paragraphs. The essential goal of this section is to summarize unification of control, communication and computation functions (in one dimensional systems) through the common thread of one dimensional neural networks.

One Dimensional Logic Theory: One Dimensional Neural Networks One dimensional logic theory as well as logic synthesis deal with information processing logic gates and logic circuits which operate on one dimensional arrays of zeroes and ones (or more generally one dimensional arrays containing finitely many symbols). The

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operations performed by AND, OR, NOR, NAND, XOR gates have appropriate intuitive interpretation in terms of the entries of the one dimensional arrays i.e. vectors. Research in the area of artifical neural networks led to the problem whether all one dimensional logic gates can be synthesized using a single layer neural network. Chakradhar et al. provided an answer to the problem. They showed that the set of stable states of a Hopfield neural network correspond to one dimensional logic functions (CAB). Equivalently, the input and output signal states of a logic gate are related through an energy function. The outputs correspond to the stable states of neural network (which constitute the local optima of the energy function). Thus, in a well defined sense, one dimensional neural networks and logic theory are related.

One Dimensional Error Correcting Codes: One Dimensional Neural Networks In (BrB), several ways of relating the concept of neural networks and the concept of error correcting codes are presented. Specifically it is shown that, given a linear block code, a neural network can be constructed in such a way that every local maximum of the energy function corresponds to a codeword and every codeword correspond to a local maximum. Also it is shown that performing maximum likelihood decoding in a linear block error correcting code is shown to be equivalent to finding a global maximum of the energy function of certain neural network. Thus, one dimensional neural networks and error correcting codes are related.

One Dimensional Optimal Control Vectors: One Dimensional Neural Networks In dealing with the problem of storage of data in magnetic and optical recording systems, Wyner formulated an important open research problem (GoC). The problem is “Consider a Single Input, Single Output, (SISO) linear time invariant continous time system. Consider the input which is constrained to assume values between +1 and –1. Determine the optimal input signals which maximize the total output energy over a finite horizon”. This problem was solved by the author in (Rama 5) and independently by Honig et al. (HoS). In (RKB), the author formulated and solved the problem in the case of SISO discrete time, linear time invariant systems. The result in the case of discrete time SISO systems shows that the optimal control vectors over a finite horizon constitute the stable states of a Hopfield Neural Network. Thus optimal control vectors are synthesized as the local optima of energy function associated with a Hopfield neural network. The associated derivation is provided in Chapter 7. Thus, the research work summarized in the previous paragraphs shows that optimal control vectors, optimal codeword vectors and optimal logic gate outputs are synthesized as the stable states of a one dimensional neural network (not necessarily same). Hence the three research areas of control, communication and computation are unified using the common thread of neural networks. One should note that the unification is done in one dimension (one independent variable). In the following, we extend the unification to multidimensions. Particularly in the following section, the main achievement of the author (in this chapter) is discussed.

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5.3 OPTIMAL CONTROL TENSORS: MULTIDIMENSIONAL NEURAL NETWORKS In the case of one dimensional linear systems, it was shown that the state space representation of the dynamics is much better than input-output description. Specifically, state space representation naturally leads to concepts such as controllability, observability associated with the system. Unfortunately, in the case of multidimensional systems, there is no natural notion of causality. Thus system theorists introduced notions such as quarter-plane causality, halfplane causality etc by partitioning the index set for state variables. In contrast to these approaches, the author discovered and formalized (Rama 3), Tensor State Space Representation (TSSR) of CERTAIN multidimensional systems. It is discussed in (Rama 3) that this particular representation enables transferring results from one dimensional systems (with vector-matrix state space representation) to certain multidimensional systems. In summary, CERTAIN multi/infinite dimensional discrete time/index dynamical systems can be described by means of a state space description of the following form:

Tensor State Space Representation of Certain Discrete Time Systems Discrete Time Systems X( i 1,..., ir ) (n + 1) = A( i 1,..., ir ; j1,..., jr ) (n) ⊗ X( j 1,..., jr ) (n) + B( i 1,..., ir ; j 1,..., jp ) (n) ⊗ U ( j 1,..., jp ) (n), Y( l1,..., ls ) (n) = C( l1,..., ls ; j 1,..., jr ) (n) ⊗ X( j 1,..., jr ) (n) + D ( l1,..., ls ; j 1,..., jp ) (n) ⊗ U( j 1,..., jp ) (n).

(5.1)

Where ⊗ denotes inner product operation between compatible tensors (BoT). Also in (5.1), A(n) is an m dimensional tensor of order 2r (called the state coupling tensor ), X(n) is the state of the dynamical system at the discrete time index n, whereas X(n+1) is the state of the system at the discrete time index n+1. Furthermore B(n) is an m dimensional tensor of order r+p ( called the input coupling tensor ), Y(n) is an output tensor of dimension m and order s. U(n) is an m dimensional input tensor of order p (varying with the discrete time index of order p) and C(n) (called the state coupling tensor to the output dynamics) is an m-dimensional tensor of order (s + r), D(n) is the input coupling tensor to the output dynamics of dimension m and order s + p. With the above important representation of certain multidimensional systems, we formulate and solve an important problem in optimal control of certain multidimensional systems. The solution of the problem shows that the optimal control tensors are synthesized as the stable states of a multidimensional Hopfield neural network (The connection structute of m -d Hopfield neural network is a fully symmetric tensor).

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Problem Definition Find an admissible sequence of (realizable) input signal tensors, U(k ) for k ∈ { 0, 1, 2, ....} (with each component of the tensor being bounded in amplitude by unity (one) or without loss of generality be a fixed constant) i.e. Ui 1, i 2,..., ir ( k ) ≤ 1 in order to minimize the criterion  −1  kf J = 2  ∑ Yin ,..., i 1 ( k ) ⊗ Yi1,..., in ( k )   k=0 

(5.2)

subject to X (n +1) = A(n ) ⊗ X (n) + B(n ) ⊗ U (n)

(5.3)

Y (n ) = C ( n ) ⊗ X ( n ) (5.4) where A(n), B(n), C(n), D(n) are tensors arising in the system dynamics of the discrete time multi/infinite dimensional system. Furthermore, X(n) is the state tensor of the system. These tensors which arise in the system dynamics are of compatible dimensions. Without loss of generality, a multi-input, multi-output multidimensional linear system is considered. Let the impulse response tensor of the system be denoted by h(k, l). This is the discrete time version of the problem given in (GoC) for CERTAIN discrete time multidimensional systems. The open problem given in (GoC) is solved in (Rama 5).

Problem Definition The optimality condition is derived through the application of the maximum principle or equivalently, the dynamic programming principle. The application of dynamic programming enables us to derive the necessary as well as sufficient condition through the principle of optimality in some cases. Discrete time, Time Varying Linear Systems: Let U(k) , k = 0, 1, 2, ..... k f − 1 be the optimal control tensor sequence, and let X (k) , k = 0,1,2 ,..., be the state response of the linear system due to the input tensors U(k), uniquely specified by (5.3), (5.4) and the initial condition of the linear dynamical system. Then, under reasonable assumptions, discussed in the application of the discrete maximum principle (SaW), it is shown that there exists a non-trivial function satisfying δ H(Xk , Uk , λk + 1 , k ) λk = (5.5) δ Xk where the Pontryagin function/Hamiltonian is given by

H(X k , U k , λk + 1 , k ) =

−1 (C(k ) ⊗ X(k ))in ,..., i1 ⊗ (C(k ) ⊗ X(k ))i1,..., in + 2

λil ,..., i 1 (k + 1) ⊗ [ A(k ) ⊗ X (k ) + B(k ) ⊗ U (k )]

(5.6)

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Thus, the adjoint tensor equation for the problem is given by

λ ( k ) = − C jm , jm −1,..., j 1 ( k ) ⊗ Yi 1,..., ip ( k ) + Ais ,..., i 1 ( k ) ⊗ λ ( k + 1)

(5.7)

Since, the terminal state is unspecified, we have

λ ( k f ) = − C jm ,..., j 1 ( k f ) ⊗ Yi 1,..., ip ( k f )

(5.8)

This will provide the terminal condition for solving (5.7). Since the input tensor sequence is constrained, it must necessarily satisfy

H (X k ,U k , λk + 1 , k ) = Min H ( Xk ,V , λk + 1 , k ) for all k = 0,1,..., k f − 1, V ∈T Where

T is the constraint set.

Thus, U ( k ) = − Sign (Bsl ,..., S1 (k ) ⊗ λ t 1 ,...., tn ( k + 1))

(5.9)

Solving (5.7) for λ(k + 1) and substituting in (5.9), we arrive at the optimal control sequence. When the constraint set is other than a hypercube, various well known techniques from mathematical programming for different constraint sets such as a convex polytope, convex polyhedra are invoked in the context of quadratic programming. The cost function is quadratic and it is optimized over various types of constraint sets such as the one described previously. With the terminal state specified, the equation (5.7) is recursed backwards to arrive at the optimal control tensor in the case of multi/infinite dimensional systems. Thus, an efficient computational form for solving the two point boundary value problem is derived in the following. It should be noted that, we derive the expression for λk +1 in the case of certain linear time varying multi/infinite dimensional dynamical systems

λ ( k ) = − C jm ,..., j 1 ( k ) ⊗ Yi 1,..., ip ( k ) + Ais ,..., i 1 ( k ) ⊗ λ ( k + 1)

(5.10)

starting with the terminal condition, recursing backwards. Remark Before we proceed further, it should be reminded that the indices for tensor describing the order of the tensor are given values by the symbols that came to mind. The tensors in the above state space representation are of compatible order to ensure that inner and outer products make sense. Now, we return to the derivation. In the following, the notation ⊗ is utilized to denote the inner product (BoT) between the tensors of compatible order.

λt1,..., tl (k f ) = – C jm ,..., j 1 (k f ) ⊗ Yi 1,..., ip (k f )

(5.11)

λt1,..., tl (k f −1 ) = − C jm ,..., j 1 (k f −1 ) ⊗ Yi 1,..., ip (k f −1 ) − Ais ,..., i 1 (k f −1 ) ⊗ C jm ,..., j 1 (k f ) ⊗ Yi 1,..., ip (k f ) (5.12)

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λt1,..., tl (k f − 2 ) = − C jm ,..., j 1 (k f − 2 ) ⊗ Yi 1,..., ip (k f − 2 ) − Ais ,..., i 1 (k f − 2 ) ⊗ C jm,..., j 1 (k f −1 ) ⊗ Yi 1,..., ip (k f − 1 ) − Ais ,..., i 1 (k f − 2 ) ⊗ Ais ,..., i 1 (k f − 1 ) ⊗ C jm,..., j 1 (k f ) ⊗ Yi 1,..., ip (k f )

(5.13)

λt1,..., tl ( k f − 3 ) = − Cjm ,..., j 1 ( k f − 3 ) ⊗ Yi 1,..., ip ( k f − 3 ) − Ais ,..., i 1 ( k f − 3 ) ⊗ Cjm,..., j 1 ( k f − 2 ) ⊗ Yi 1,..., ip (k f − 2 ) − Ais ,..., i 1 (k f − 3 ) ⊗ Ais ,..., i 1 ( k f − 2 ) ⊗ C jm ,..., j 1 (k f − 1 ) ⊗ Yi 1,..., ip ( k f − 1 ) – A is ,..., i 1( k f – 3 ) ⊗ Ais ,..., i 1 ( k f – 2 ) ⊗ Ais ,..., i 1 ( k f –1 ) ⊗ C jm ,..., j 1 ( k f ) ⊗ Yi 1,...,ip ( k f )

(5.14)

Thus, continuing the solution of the difference equation backwards, we have

λt1,..., tl ( k f − l ) = − Cjm ,..., j 1 ( k f − l ) ⊗ Yi 1,..., ip ( k f − l ) − Ais ,..., i 1 ( k f − l ) ⊗ Cjm,..., j 1 ( k f − l + 1 ) ⊗ Yi 1,..., ip (k f – l + l ) − Ais ,..., i 1 (k f – l ) ⊗ Ais ,..., i 1 (k f – l + 1 ) ⊗ C jm ,..., j 1 (k f – l + 2 ) ⊗Yi 1,..., ip (k f – l + 2 ) –... – A is ,...,i 1 ( k f – l ) ⊗ A is ,..., i 1 ( k f – l + 1 ) ⊗ ... ⊗ A is ,..., i 1 ( k f – 1 ) ⊗ C jm ,..., j 1 ( k f ) ⊗ Yi 1,..., (k f )

(5.15)

Let l = k f − k −1 . This implies k f − l = k +1 . Hence, by the substitution,

λtl ,..., tl ( k + 1) = − C jm ,..., j 1 ( k + 1) ⊗ Yi 1,..., ip ( k + 1) − Ais ,..., i 1 ( k + 1) ⊗ Cjm,..., j 1 ( k + 2) ⊗ Yi 1,..., ip (k + 2) − Ais ,..., i 1 (k + 1) ⊗ Ais ,..., i 1 ( k + 2) ⊗ C jm,..., j 1 (k + 3) ⊗ Yi 1,..., ip (k + 3) −... − Ais ,..., i 1 ( k + 1) ⊗ Ais ,..., i 1 (k + 2) ⊗ ... ⊗ Ais ,..., i 1 ( k + l) ⊗ C jm ,..., j 1 (k + l + 1) ⊗ Yi 1,..., ip ( k + l + 1) (5.16) Thus we have the optimal control solution for the problem given by (utilizing (5.9))

Uv 1,..., vr ( k ) = Sign (Bsl ,..., s1 (k ) ⊗ C jm ,..., j 1 (k + 1) ⊗ Y( k + 1) + l

∑B i =1

sl ,..., s1

( k ) ⊗ Ais ,..., i 1 (k + 1) ⊗ .... ⊗ Ais ,..., i 1 ( k + i) ⊗ C jm ,..., j 1 ( k + i + 1) ⊗ Yi 1,..., ip (k + i + 1) )

(5.17)

Now, utilizing the definition of the impulse response tensor of the time varying linear system, we have

Uv 1,..., vr ( k ) = Sign (Bsl ,..., s1 ( k ) ⊗ C jm ,..., j 1 ( k + 1) ⊗ Y( k + 1) + l

∑ h (k + i + 1, k ) ⊗ Y(k + i + 1)) i =1

l

= Sign (∑ h (k + i + 1, k ) ⊗ Y(k + i + 1) ) i=0

(5.18)

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h(.,.) is the transposed tensor of the impulse response tensor. The term in the parenthesis is given by l

l

k +i +1

i=0

i=0

j=0

∑ h (k + i + 1, k) ⊗ Y(k + i + 1) = ∑ h (k + i + 1, k ) ⊗ ∑ h (k + i + 1, j) ⊗ u ( j)

(5.19)

Exchanging the order of summation, (with the help of associated index grid), we have kf

kf − k − 1

j =0

i = max imum {0, j − k − 1}

∑

∑

( h ( k + i + 1, k ) ⊗ h ( k + i + 1, j) ⊗ u( j)

(5.20)

Rewriting the above expression for optimal control, we have kf − k − 1  kf  U ∗ (k ) = Sign  ∑ (h (k + i + 1, k ) ⊗ h (k + i + 1, j )) ⊗ U ( j )  ∑  j = 0 i = max{1, j − k − 1} 

(5.21)

Let us define kf − k − 1

∑

R(k , j ) =

i = max{1, j − k − 1}

(h (k + i + 1, k ) ⊗ h (k + i + 1, j))

(5.22)

Thus, we have for the optimal control, kf

U ∗ ( k ) = Sign ( ∑ R( k , j) ⊗ U( j)) j=0

(5.23)

Now, for the time invariant linear systems, we have

R( k , j) =

kf − k − 1

∑ (h(i + 1) ⊗ h(k + i + 1 − j)

i = max {1, j − k − 1}

(5.24)

This is the energy density tensor of time invariant linear system obtained from the impulse response tensor. Thus the optimal control tensor is the stable state of a multidimensional Hopfield neural network. Continuous Time Dynamical Systems Now, we formulate and solve the continuous time versions of the problems. The continuous time versions of the problem provides us with the structure of the local optimum of a quadratic form over the continuous time multi/infinite dimensional hypercube. This is the problem where the L∞ norm of the control tensors is constrained in amplitude by unity. In the derivation of the optimal control, the following definition is necessary. Integral of Tensor Function of a Scalar Argument: By the integral of a tensor of a scalar continuous argument, we mean the tensor with the components,

Unified Theory of Control, Communication and Computation: Multidimensional Neural Networks

∫A

i 1,..., i

(t) dt or

∫A

i 1,..., in

(t1 ,..., tn ) dt1 dt 2 ... dm

87

(5.25)

Optimal Control Problem Formulation Consider a multi/infinite dimensional linear system with continuous index/argument. The system dynamics are given by . X i 1,..., ir ( t ) = Ai 1,..., ir ; j 1,..., jr (t) ⊗ X j 1,..., jr (t ) + Bi 1,..., ir ; j 1,..., jp (t ) ⊗ U j 1,..., jp (t)

Yl 1,..., ls (t) = Cl1,..., ls ; j 1,..., jr (t) ⊗ X j 1,..., jr (t)

(5.26)

The objective function being minimized in the optimal control problem is given by tf

tf

−1 J = ∫ Yls ,..., l 1 (t) ⊗ Yl 1,..., ls (t) dt = ∫ φ (X , U , t ) dt 2 to to

(5.27)

subject to the constraint given in (5.26) and the input tensors are constrained to be on the continuous time multi/infinite dimensional hypercube. Solution: Form the Pontryagin function ( or Hamiltonian) of the problem. It is given by H (X , U , λ , t) =

−1 (C(t) ⊗ X(t))ls ,..., l 1 ⊗ (C(t) ⊗ X(t))l 1,..., ls + 2 λir ,..., i 1 (t) ⊗ ( A(t) ⊗ X(t) + B(t) ⊗ U(t))

(5.28)

Minimize the Pontryagin function H ( X , U , λ , t ) with respect to all the admissible control tensors i.e. control tensors whose components are constrained in amplitude by unity. Thus,

U*

j 1,..., jp

(t) = − Sign {Bjp ,..., j 1;ir ,..., i 1 (t) ⊗ λi 1,..., ir (t)}

Thus, the optimal control tensors for the problem is obtained from the above equation. To explicitly determine the optimal control, the adjoint equations and associated boundary conditions are given by

. δ H(X , U , λ , t) δφ ( X , U , t) δ T − λi 1,..., ir (t) = = + [ A(t) X(t) + B(t)U(t)] ⊗ λ(t), δX δX δX where

δ is a partial derivative operator, δX

(5.29)

λi 1,..., ir (t f ) = 0 The above equations (5.28), (5.29) alongwith the system dynamics described through (5.26) are solved for determining λi 1,..., ir (t)

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i −λi 1,..., ir (t) = − Cls ,..., l1 (t) ⊗ Cl1,..., ls (t) ⊗ Xi 1,..., ir (t)  + Ajr ,..., j 1; ir ,..., i 1 (t) ⊗ λi 1,..., ir (t) with λi 1,..., ir (t f ) = 0

(5.30)

λi 1,..., ir (t) = − Ajr ,..., j 1; ir ,..., i 1 (t) ⊗ λi 1,..., ir (t) + Cl 1,..., ls (t) ⊗ Yl 1,..., ls (t) with λi 1,..., ir (t f ) = 0 .

(5.31)

The above differential equation is solved, like the state equations for the linear dynamical system, to arrive at t

a a λi 1,..., ir (t) = φ (t , t f ) ⊗ λ (t f ) + ∫ φ (t , τ ) ⊗ Cls ,..., l 1 (τ ) ⊗ Yl 1,..., ls (τ ) dτ , tf

d a φ (t ,τ ) = − Ajr ,..., j 1; ir ,..., i 1 (t) ⊗ φ a (t , τ ) dt

(5.32)

(5.33)

φ a ( t ,τ ) = I ; φ a ( t , t f ) = φ ( t f , t ) a where φ (t f , t ) is the state transition tensor. Thus, we have

t

a λi 1,..., ir (t) = ∫ φ (t ,τ ) ⊗ Cls ,..., l1 (τ ) ⊗ Yl1,..., ls (τ )dτ tf

t

=

∫ φ (τ , t) ⊗ C

ls ,..., l1

(τ ) ⊗ Yl1,..., ls (τ )dτ

tf

(5.34)

tf

= − ∫ φ (τ , t) ⊗ Cls ,..., l1 (τ ) ⊗ Yl1,..., ls (τ )dτ t

Hence, we have

−Bjp ,..., j 1; ir ,..., i 1 (t) ⊗ λi 1,..., ir (t) = tf

∫B

jr ,..., j1; ir ,..., j1

(t) ⊗ φ (τ , t) ⊗ Cls ,..., l 1 (τ ) ⊗ Yl 1,..., ls (τ )dτ

(5.35)

t

where τ

Yl1,..., ls (τ ) = ∫ Cl1,..., ls (τ ) ⊗ φ (τ , s) ⊗ B( s) ⊗ U ( s)ds to

(5.36)

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Thus, we have − B jp , ..., j 1 ; ir , ..., i 1 (t) ⊗ λ i1, ..., ir (t) tf

=

∫

τ

B j p , ..., j1 ; ir , ..., i1 (t ) ⊗ φ (τ , t ) ⊗ C ls, ..., l1 (τ ) ⊗ [

t

∫

C(τ ) ⊗ φ (τ , s) ⊗ B(s ) ⊗ U ( s ) ds] dτ

(5.37)

t0

Exchanging the order of integration (with associated index diagram)

−Bjp ,..., j 1; ir ,..., i 1 t ⊗ λi 1,..., ir (t) = tf tf

∫ ∫B

jp ,..., i 1

(t) ⊗ φ (τ , t) ⊗ Cls ,..., l 1 (τ ) ⊗ Cl1,..., ls (τ ) ⊗ φ (τ , s) ⊗ Bi 1,..., jp ( s) dsdτ

(5.38)

0 s

But, the impulse response tensor is given by

 H j1 ,..., jp (t , τ ) = Cl 1,..., ls (t) ⊗ φ (t , τ ) ⊗ Bi 1,..., jp (τ )  t <τ 0

t ≥τ

(5.39)

Utilizing the above expression in (5.37), tf tf

− B jp ,..., j 1; ir ,..., i 1 ( t ) ⊗ λ j1 ,..., ir ( t ) = ∫ ∫ H jp ,..., j1 (τ , t ) ⊗ H j1 ,..., jp (τ , s) dτ ⊗ U ( s) ds 0 s

(5.40)

Thus, the optimal control tensor is given by

U * j 1,..., jp (t) = −Sign (Bjp ,..., j 1) (t) ⊗ λj 1,..., jr (t)  tf  = Sign  ∫ R(t , s) ⊗ U * j 1,..., jp ( s)ds    0 

(5.41)

where R(t,s) is the energy density tensor of the linear system and is given by tf

R(t , s) = ∫ H jp ,..., j1 (τ , t ) ⊗ H j1 ,..., jp (τ , s) dτ

(5.42)

s

For, linear time invariant multidimensional systems, H (τ , s ) , the impulse response tensor is dependent only on the difference between arguments/indices. Thus, the necessary condition on the optimal control (for continuous time multidimensional systems) is given by (5.41). It shows that the optimal control tensor is the stable state of a continuous time (Hopfield type) neural network. One must understand that the concept of continuous time multidimensional neural network is conceived by the author in (Rama 4). It should be noted that when the objective function is a higher degree form (rather than quadratic form), similar derivations are done. Details are avoided for brevity.

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5.4 MULTIDIMENSIONAL SYSTEMS: OPTIMAL CONTROL TENSORS, CODEWORD TENSORS AND SWITCHING FUNCTION TENSORS In view of the results in previous section, in the following, we briefly summarize the results discussed in (Rama 1) and (Rama 2) so that we realize that the unification (of control, communication and computation functions) in one dimension, discussed and formalized by the author also naturally extends to multidimensional systems.

Multidimensional Logic Theory: Multidimensional Neural Networks In one dimensional logic theory, operations performed by AND, OR, NOR, NAND, XOR, NOT gates have appropriate intuitive interpretation in terms of the entries of the one dimensional arrays i.e.vectors. Any effort to generalize the one dimensional logic operations to multidimensions leads to various heuristic possibilities and requires considerable ingenuity in formalizing the definition. But, in the following, utilizing the multidimensional neural network model [ described in (Rama1) ], a formal/mathematical procedure to multidimensional logic theory is described. The input and output signal states of a multidimensional logic gate are related through an energy function. Equivalently, the multidimensional logic functions are associated with the local optima of various energy functions defined over the set of input multidimensional arrays. In view of the mathematical model of multidimensional neural network [ conceived in (Rama1) ], it is most logical to define the maximum/minimum energy states of a multidimensional neural network (optimizing an energy function over the multidimensional hypercube) to correspond to the multidimensional logic gate functions operating on the input arrays. Similarly generalized multidimensional neural networks (optimizing a higher degree form) are utilized to define generalized logic functions. Now we summarize the relationship between multidimensional codes and multidimensional neural networks.

Multidimensional Coding Theory: Multidimensional Neural Networks In the field of one dimensional error correcting codes, various linear as well as non-linear codes are designed by various researchers. The approaches utilized were mathematically very sound. But some researchers tried to extend the approach to two/three/ multidimensional code design with limited success. The author (for the first time) conceived the idea of utilizing a generator tensor to represent a multidimensional linear code. Given this idea and the results in (BrB) [ relating one dimensional linear/non-linear codes to generalized neural networks], multidimensional linear as well as non-linear codes are related to multidimensional generalized neural networks. One specific result is the following (Details can be found in (Rama 2) ): Given a multidimensional block code (linear or non-linear), a neural network can be constructed in

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such a way that every local maximum of the energy function corresponds to a codeword tensor and every codeword tensor corresponds to a local maximum (i.e. stable state). Unification: Now utilizing the results in Section 3 (relating optimal control tensors and multidimensional neural networks), we readily have the unification of control, communication and computation (through the common thread of neural networks). Formally, the optimal control tensors, optimal multidimensional logic functions, multidimensional codeword tensors are synthesized through the stable states of multidimensional neural (generalized) networks. In the above unification discussion, we only considered neural (generalized neural) networks in discrete time. In equation (5.41), we discovered and formalized the concept of continuous time neural associative memory (with the energy function being a quadratic form associated with certain Kernel). Continuous time generalized neural networks are defined and associated with optimal control tensors, optimal codeword tensors and optimal switching functions. Unified theory with generalized neural networks follows in a similar fashion. Details are avoided here for brevity. In view of formal clarity, the following theorem is a comprehensive statement of the unification of control, communication and computation functions (with quadratic energy function/objective function) The generalization to the case of higher degree energy function follows in a similar manner. Theorem 5.1: Consider a linear time varying multidimensional system with the state space representation provided in (5.3), (5.4). The optimal control tensor (subject to a finite amplitude constraint, i.e. Uv1,..., vr ≤ 1 o r, N), optimal switching function (in the sense of a transformation between an input tensor and an output tensor), optimal linear multidimensional code constitute the local optimum of a quadratic form in the components of state variable, input, output tensors. Thus, in the case of linear dynamical systems, with quadratic energy/objective function, the optimal control tensors, optimal switching function, optimal linear code are unified to be the local optima of a quadratic form (with argument/index/time varying coefficient tensors for time varying systems) over the multidimensional hypercube. Thus these local optima are synthesized as the stable states of neural/generalized neural network. Proof: From (Rama1), the stable states of a multidimensional neural network constitute the local optimum of a quadratic form with the fully symmetric connection tensor as the weighting tensor. The convergence theorem for (infinite) multidimensional neural networks provides a formal result. These local optima are defined to be the multidimensional logic functions in the sense of a mapping between the input tensors and the stable state tensor. But, from (5.23), the optimal control tensors which optimize a quadratic objective function have the stable state structure of an interconnected multidimensional neural network with

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block fully symmetric connection structure. Thus, the optimal control and optimal switching function which optimize a quadratic objective function constitute the stable states of a multidimensional neural network. From (Rama 2), it is formally true that the connection structure of a multidimensional graph-type structure (say graphoid) is associated with a multidimensional linear code through its cut space. These cutset codes are termed graph-theoretic codes. It is also proved in (Rama 2) that maximum likelihood decoding of a corrupted word (received word) with respect to the graphoid theoretic code is equivalent to finding the global optimum of the quadratic energy function associated with a multidimensional neural network. Furthermore, it is shown that a tensor constitutes the local optimum of a multi-variate polynomial in the components of input, output tensors (quadratic tensor form) if and only if ( the polynomial is associated with the parity check tensor) it is a codeword of the multidimensional linear code. Thus, associated with the generator/parity check tensor of graphoid theoretic code, there exists a quadratic form whose local optimum constitute the codewords (quadratic form over the multidimensional hypercube). Hence the optimal code, optimal control and optimal switching function which constitute the local optimum of a multi-variate quadratic form ( in the components of state, input, ouput tensors) are unified to be the same. This constitutes the statement of unified theory of control, communication and computation in linear dynamical systems (time varying as well as time invariant systems) with a quadratic form as the objective function. Q. E. D. In a future revision, it is discussed how the unification extends to other important functions. Generalization of the results to certain infinite dimensional systems is also discussed.

5.5 CONCLUSIONS In this chapter, based on the work of author and earlier authors, the unification of control, communication and computation functions (through the common thread of neural networks) is formalized. The main contribution of the author for unification in one dimension is to show that the optimal control vectors (in a well known optimality criterion) constitute the stable states of a Hopfield network. The next important step was to envision unification in multidimensions. Based on the concept of multidimensional neural networks (Rama1, Rama 2), the author was able to formally unify communication and computation functions. Tensor State Space Representation (TSSR) conceived and formalized by the author was utilized to prove that the optimal control tensors constitute the stable states of a multidimensional neural network (in discrete as well as continuous time systems). With this important result, the author was able to show that optimal codewords, optimal logic functions and optimal control tensors constitute the stable states of a multidimensional neural network.

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REFERENCES (BoT) A. I. Borisenko and I. E. Tarapov, “Vector and Tensor Analysis with Applications“, Dover Publications Inc., New York, 1968. (BrB) J. Bruck and M. Blaum, “Neural Networks, Error Correcting Codes and Polynomials Over the Binary Hypercube“, IEEE Transactions on Information Theory, Vol. 35, No. 5, September 1989. (CAB) S.T. Chakradhar, V.D. Agrawal and M.L. Bushnell, “Neural Models and Algorithms for Digital Testing“, Kluwer Academic Publishers, 1991. (GoC) B. Gopinath and T. Cover, “Open Problems in Control, Communication and Computation“, Springer, Heidelberg, 1987. (HoS) M.Honig and K. Stieglitz, “On Wyner’s conjecture” Bellcore Technical Memorandum. (Rama 1) Garimella Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory“, International Journal of Neural Systems, Vol.15, No.3, Pages 223-235, June 2005. (Rama 2) Garimella Rama Murthy, “Multidimensional Coding Theory: Multidimensional Neural Networks“, In part presented at the 2002 IEEE International Workshop on Information Theory. (Rama 3) G. Rama Murthy, “Tensor State Space Representation: Multidimensional Systems“, International Journal of Systemics, Cybernetics and Informatics (IJSCI), January 2007, pages 16-23. (Rama 4) G. Rama Murthy, “Optimal Control, Codeword, Logic Function Tensors: Multidimensional Neural Networks”, International Journal... (IJSCI), October 2006, Pages 917. (Rama 5) G. Rama Murthy, “Signal Design for Magnetic and Optical Recording Channels: Spectra of Bounded Functions“, Bellcore Technical Memorandum, TM-NWT-018026, December 1990. (RKB) G. Rama Murthy, P. Krishna Reddy and L. Behera, “Neural Network Based Optimal Binary Filters”, submitted to Elsevier Signal Processing Journal. (Gop) M. Gopal, “Modern Control System Theory“, John Wiley and Sons, New York, (SaW) Sage and White, “Optimum Systems Control“, Prentice-Hall Inc., Englewood Cliffs, New Jersey 07632.

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CHAPTER

6

Comple xV alued Neural Complex Valued Associative Memory on the Comple x Hypercube Complex

6.1 INTRODUCTION The Hopfield model of the neural network is designed basing on the McCulloch-Pitts neuron. In this network the computation of the algebraic threshold function is carried out at each node. The edge between two nodes is associated with a weight. This network can hence be represented with a weight matrix which is nothing but a symmetric matrix where

Wi , j represents the weight associated with the edge connecting the neurons i and j . Since it is a symmetric matrix, (i.e., the network represented by an undirected graph), we have

Wi , j = W j ,i . The threshold function can be calculated at each neuron using the function  1, if Hi (t ) ≥ 0 Vi (t + 1) = sgn ( Hi (t) ) =  −1 otherwise where, n

Hi (t ) = ∑ Wj , i Vj (t ) − Ti j =1

Here, Vi (t + 1) represents the value of the function i.e. state value at node i at time (t +1) (which is the next time instant). Energy function: The model also associates an energy function which is the quadratic form V T (t )WV (t ) (neglecting the threshold value without loss of generality) where V(t) stands for the column matrix that represents the vector corresponding to the state of all neurons at time instant t. This vector will lie on the hypercube whose order is that of the synaptic weight matrix.

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Multidimensional Neural Networks: Unified Theory

Modes of operation: The Hopfield model can operate in one of the two modes, serial or fully parallel mode or a combination of these. Serial mode is the one in which the next state computation, i.e., the evaluation of the neural network takes place at each node (node after node) for every time instant. In the fully parallel mode the evaluation takes place for every node at each time instant. A combination implies that the evaluation occurs at a group of nodes for every time instant. A stable state is defined as a state such that after reaching it, the network output does not change i.e., V(t) = sgn(WV(t)). The model results in the following convergence theorems: Theorem 1: If the neural network is operating in the serial mode and the elements on the diagonal of connection matrix are non-negative, the network will converge to a stable state i.e., there are no cycles in the state space. Theorem 2 : If the network is operating in the fully parallel mode, the network will either converge to a stable state or to a cycle of length two i.e., it oscillates between two states in the state space. Goals: The goals of this chapter are to consider the possibilities of implementing a complex valued associative memory and observe the behavior of the model in the serial and the fully parallel modes. Remark Recently the concept of a complex valued neural network has been explored since the work of [ZURADA 1996] and has been almost successfully applied to the fields of image processing and pattern recognition. A conglomeration of the papers on the subject has been briefly collected in [HIROSE 2003]. Following this literature our work is based on implementation of a newer method to realize a complex valued neural network. The chapter is organized into three parts. The first part of section 2 discusses the features of the model the authors are proposing. Also implications to convergence of the network are briefly pointed out. The second part of the same section provides a proof technique used for arguing the convergence properties of the discussed form of the complex valued associative memory. The third part actually presents the proof of convergence and considers how it is similar to real valued Hopfield associative memory.

6.2 FEATURES OF THE PROPOSED MODEL The model that we are about to propose considers a case where the neuron output is complex valued and lies on the complex hypercube. A vector of size n on the complex hypercube has each entry element belonging to the set {1 + j , 1– j , – 1 + j , -1 – j}.Thus, the state vector also lies on the complex hypercube. This is, in many ways similar to the Hopfield model.

Complex Valued Neural Associative Memory on the Complex Hypercube

97

The Synaptic weights are complex-valued and the weight matrix is Hermitian unlike the real valued case, where it is symmetric. The next state V(t+1) can be computed as, V(t+1) = sgn (real part (WV(t))) + jsgn (complex part (WV(t)))

(6.1)

Thus the values of the entities in the column vector V(t+1) anytime would be confining to the set {1+J, 1–j, –1+j, –1–j} unlike the real case wherein the values confine only to the set {1, –1}. Thus the total number of values V(t+1) would take, i.e., the number of points of the “Complex hypercube” would equal 4n where n is the order of the neural network. The energy function would thus be E(t) = (VT(t))* WV(t) (neglecting the threshold value without loss of generality). The authors would like to prove that an important property of this model would be to converge to a stable state when operating in the serial mode and utmost to a cycle of length 2 when operating in the fully parallel mode. The proof technique adopted by the authors is method of isolating the real and imaginary parts of the Hermitian synaptic weight matrix and evaluating them separately. As one can see when the Hermitian matrix is isolated into two parts real and imaginary, the matrix corresponding the real part would be a real symmetric one and that corresponding the imaginary part would be a real anti-symmetric one. Remark It is an interesting observation that the energy function when evaluated for the real part with the complex valued vector would behave exactly as if it were a matrix being evaluated for the real valued neural network proposed in [HOPFIELD].That is, we have complex valued associative memory with a real connection matrix. The exact details of the proof follow.

6.3 CONVERGENCE THEOREMS Convergence Issues in Different Modes of Operation Before delving into the proof we summarize convergence issues. 1. Given a neural network N = [W,T] with the synaptic weights in the weight matrix being complex and the matrix itself being Hermitian and the calculation of the algebraic threshold function generating a complex number, the network always converges to a stable state when operating in the serial mode. 2. The same network when operating in the parallel mode either converges to a stable state or to a cycle of length (maximum) two. Generalized proof of convergence of an arbitrary neural net in the serial mode. From the proof technique discussed in section (II) the proof follows. Ek (t) is the evaluation of the energy function value at node k at time instant t.

Multidimensional Neural Networks: Unified Theory

98

Ek (t ) = (V (t)" Vk (t ) " V (t ) ) * 1

*

* n

 W11   # W  n1

 V1 (t )    " W1 n   #    O #   Vk (t )   " W nn   #    Vn (t ) 

If we break the expression for Ek (t) into two parts, EkR(t) and Ek i(t), the real and imaginary parts (of energy function), they come out like this:

(

EkR ( t ) = V1 (t)"Vk (t) "Vn (t) *

*

*

)

 W11R " W1nR    " #   # W   1nR " WnnR 

 V1 (t )     #   V (t )   k   #   V (t )   n 

" jW1nI   0   # " #  * * * EkI (t ) = (V1 (t )"Vk (t ) "Vn (t ) )   − jW " 0  1nI 

(6.2)

 V1 (t )     #   V (t )   k   #   V (t )   n 

Evaluating the real part of (6.1) for the energy function, we have,

 Vj ( t ) + i=0 j= k +1  j =1  k n 1 −   Ek (t ) = Vk * (t )  ∑ WkjRVj ( t ) + WkkRVk ( t ) + ∑ WkjRVj ( t ) + j= k +1  j=1  n n  k −1  Vi * (t ) ∑ WijRVj (t ) + WikRVk ( t ) + ∑ WijRVj ( t ) ∑ i =k +1 j =k +1  j =1  k −1

∑V

i

*

 k −1

( t ) ∑ WijRVj ( t ) + WikRVk ( t ) +

n

∑W

ijR

Similarly,

 W11R  EkR (t + 1) = (V1 (t )" Vk (t + 1)"Vn (t ))  # W  n1 R *

*

*

 V1 (t )    " W1nR   #   #  Vk (t + 1)  O   " WnnR   #   V (t )   n 

(6.3)

Complex Valued Neural Associative Memory on the Complex Hypercube

99

The expression for Ek R(t+1) results because it is operating in the serial mode and the updating of the function value takes place at only one node, i.e., the node at which we are evaluating(Vk ). In the parallel mode, however, all the function values in the vector will be updated. n  k −1  * V t W V t W V t WijRVj (t ) + 1 + + + ( ) ( ) ( )  ∑ ∑ ∑ i ijR j ikR k i=0 j= k +1  j =1  k 1 n −   Vk * (t + 1)  ∑ WkjRVj (t ) + WkkRVk ( t + 1) + ∑ WkjRVj ( t ) + j =k +1 EkR ( t + 1)  j =1  = n n  k −1  Vi * (t ) ∑ WijRVj (t ) + WikRVk ( t + 1) + ∑ WijRVj (t ) ∑ i =k +1 j =k +1  j =1  k −1

∆EkR ( t ) =

WkkR (Vk* ( t + 1)Vk (t + 1) − Vk* ( t )Vk ( t )) + n

∑

j = 1( j≠ k )

(

n

) ∑

WkjR Vk* ( t + 1)Vj (t ) − Vk* (t )Vj (t ) +

j = 1( j≠k )

(

WjkR Vk (t + 1)Vj* ( t ) − Vk (t )Vj* (t )

But since the real part of both matrices are symmetric, WkjR = WjkR ; Thus,

∆EkR ( t ) =

WkkR (Vk* (t + 1)Vk (t + 1) − Vk* (t )Vk (t )) + n

∑

j = 1( j≠ k )

∆EkR ( t ) =

(

)

)

WkkR (VkR2 ( t + 1) + VkI2 (t + 1)) − (VkR2 (t ) + VkI2 (t )) + n

∑

j = 1( j≠ k )

∆EkR ( t ) =

(

WkjR Vk* ( t + 1)Vj ( t ) − Vk* ( t )Vj ( t ) + Vk (t + 1)Vj* ( t ) − Vk ( t )Vj* ( t )

(

WkjR Vj ( t ) ∆Vk* ( t ) + Vj* (t ) ∆Vk (t )

(

)

)

WkkR ∆VkR (t ) (VkR (t + 1) + VkR (t )) + ∆VkI (t )(VkI (t + 1) + VkI (t )) + n

∑

j = 1( j≠ k )

(

WkjR Vj (t ) ∆VkR ( t ) − jVj ( t ) ∆VkI ( t ) + Vj* ( t ) ∆VkR ( t ) + jVj* (t ) ∆VkI ( t )

)

)

Multidimensional Neural Networks: Unified Theory

100

Thus on cancellation of the corresponding imaginary parts

∆EkR (t ) =

(

)

WkkR ∆VkR ( t ) (VkR (t + 1) + VkR ( t )) + ∆VkI (t )(VkI ( t + 1) + VkI (t )) + n

∑

j = 1( j≠ k )

(

)

2WkjR VjR (t ) ∆VkR (t ) + VjI (t ) ∆VkI (t )

On adding and subtracting the following term,

WkkR ∆VkR ( t )VkR (t ) + WkkR ∆VkI (t )VkI (t ) we get, n   2  WkkRVkR ( t ) + ∑ WkjRVjR (t )  ∆VkR ( t ) + WkkR ∆VkR2 ( t ) +   j =1( j ≠k )   ∆EkR (t ) = n   2  WkkRVkI (t ) + ∑ WkjRVjI (t )  ∆VkI (t ) + WkkR ∆VkI2 (t )   j =1( j≠ k )  

(6.4)

n n     If we consider that  W k k RVkR (t ) + ∑ W k jRVjR (t )  and  W k k RVk R (t ) + ∑ W k jRV jR (t )  j =1( j ≠k ) j =1( j ≠k )    

are expressions for H k (t) in the real mode with some arbitrary VjR (t) and VjI (t), then from the Hopfield convergence theorem, it is proved from the expression for

∆E = 2 Hk ∆Vk2 (t) + Wkk ∆Vk2 (t) , that it is a value that eventually goes to zero which means that Ek (t) is not which is the local maxima. Hence ∆EkR (t) in complex case also reaches zero hence Ek (t) is also not-to a local maxima. Thus it remains to evaluate the imaginary part contribution of energy function i.e.

 0  EkI (t ) = (V1 ( t )"Vk (t )"Vn ( t ))  #  −W 1 nI  *

*

*

 V1 ( t )    " W1nI   #   #   Vk ( t )  O   " 0   #     Vn (t ) 

Complex Valued Neural Associative Memory on the Complex Hypercube k −1





k −1

∑  V (t ) ∑ i =1



i

*

 j =1( i ≠ j )

jWijI Vj (t ) + jWikIVk (t ) +

101

 jWijIVj (t )   +  j = k + 1( i ≠ j )  n

∑

n  k −1  * EkI (t ) = Vk (t ) ∑ jWijIVj (t ) + ∑ jWijI Vj (t )  + j = k +1  j =1  n  n  k −1  *  Vi (t )  ∑ jWijI Vj (t ) + jWikIVk (t ) + ∑ jWijIVj (t )   ∑   i =k +1  j = k + 1( i ≠ j )  j =1(i ≠ j ) 

and similarly,

 0  EkI ( t + 1) = (V1 (t )"Vk ( t + 1)"Vn ( t ))  #   − jW1nI *

*

k −1



∑  V i =1



i

*

*



(t )

k −1

∑

 j =1( i ≠ j )

" " "

 V1 (t )    #  jW1nI    #   Vk (t + 1)    0   #     Vn (t ) 

jWijI Vj (t ) + jWikIVk (t + 1) +

 jWijI Vj (t )   +  j = k + 1( i ≠ j )  n

∑

n  k −1  EkI ( t + 1) = ∆ Vk * (t + 1) ∑ jWijIVj (t ) + ∑ jWijI Vj (t )  +  j =1  j = k +1   n n   k −1  *  Vi (t ) ∑ jWijI Vj (t ) + jWikIVk (t + 1) + ∑ jWijI Vj (t )   ∑ i =k +1  j = k + 1( i ≠ j )  j =1(i ≠ j ) 

Now since Wij = −Wji , ∆EkI (t ) = 2

n

∑ (W

j =1( j ≠ k )

kjI

) (

)

VjR ( t ) ∆VkI − WkjI VjI (t ) ∆VkR 

(6.5)

Thus, n  2 ( HkA ) ∆VkR ( t ) + WkkR ∆VkR2 (t ) +  ∆Ek (t) =   + 2 ∑  WkjI VjR (t ) ∆VkI − WkjI VjI ( t ) ∆VkR  2 j =1( j ≠ k )  2 ( HkB ) ∆VkI (t ) + WkkR ∆VkI (t ) 

(

) (

)

(6.6) This is the expression for ∆Ek (t) in the complex valued neural network.

Multidimensional Neural Networks: Unified Theory

102

As one can observe from the above expression, the first term is zero when the neural net converges to a stable state(from [BRUCK 1987]). The second term is real but may take negative values depending on the imaginary parts of the corresponding entities of the weight matrix. But when the first term becomes zero, i.e., when the net converges to a stable state, ∆VkI will be zero. Hence the second term will be zero at the stable state. Which means that the energy function of a complex valued neural net converges to a positive value. This also proves that the complex valued associative memory constrained with a real connection matrix converges to a stable state with a behavior that matches the real valued neural net described by Hopfield. Graphs of convergence of the energy function to a stable value and that of the entity ∆E to zero are depicted below. The relative performance as well as the analogous relationship of that of three cases depicted (i) complex synaptic weight matrix and complex vector, (ii) convergence for real valued neural network and (iii) an intermediary of these two, i.e., a test with complex vectors on a real valued synaptic weight matrix is shown. Convergence of Energy function 800 700

Complex weights Complex vector

500

Real weights complex vectors

400 300 200

Real weights real vectors

100 0 -100 -200 0

1

2

3

4

5

6

7 8 time t

9

10

11

12

13

14

15

Convergence of Difference of energy to zero 300 Complex weights Complex vector 250

Energy difference

Energy function value E

600

Real weights complex vectors

200

150

100

Real weights real vectors

50

0 0

2

4

6

8 time t

10

12

14

16

Complex Valued Neural Associative Memory on the Complex Hypercube

103

Thus as we see from the figures above, in the first graph the energy function value is the greatest for the case where weights are complex and the vectors are also complex. That is because the complex part of the weight matrix is non-zero and contribute to the energy value. Next comes the case where weights are real but the vectors are complex. In this case, though there will be no complex part of the weights contributing, the complex part of the vectors contribute to this increase in the energy value from the original Hopfield case which in turn has the least local optima of convergence. It is customary and mandatory to prove that this analogue between the complex and the real cases of Hopfield associative memory in the serial mode can also be extended to work in the fully parallel mode. It has been observed however, that a separate proof is not required to illustrate the behavior of the model in the fully parallel mode. The same proof can be extended in a certain manner shown by the work of [BRUCK 1987]. Before going into the extension needed, let the general form of the expression for the fully parallel mode be observed first.

 W11R  EkR ( t + 1) = (V1 (t + 1)"Vk (t + 1)"Vn (t + 1)) # W  n 1R *

*

*

 W11I  EkI (t + 1) = (V1 (t + 1)"Vk (t + 1)"Vn (t + 1))  # W  n 1I *

*

*

 V1 (t + 1)    " W1nR   #   " #  Vk (t + 1)    " WnnR  #     Vn (t + 1)   V1 (t + 1)    " W1nI   #   " #   Vk (t + 1)    " WnnI   #     Vn (t + 1) 

These expressions change because the computation of the function is done at every node of the neural net at a certain time instant. Instead of evaluating the above expressions, an easier method is to define a special neural net N’ such that, N′ = where and

[W ‘, T ‘]

 0 W W′ =  W 0    T  T′ =  T   

Multidimensional Neural Networks: Unified Theory

104

As it can be seen from the above matrix N’, it defines a newer neural net which corresponds to a bipartite graph with 2n nodes. Let the subsets of nodes be P1 and P2 which are independent sets of nodes. It has been proved beyond doubt in previous work that 1. for any serial mode of operation in N there exists a serial mode of operation in N’ provided W has a non-negative diagonal. 2. there exists a serial mode of operation in N’ which is equivalent to a fully parallel mode of operation in N. This can be seen because if N is operating in a fully parallel mode, since P1 and P2 correspond to independent sets of nodes, it would be equivalent to evaluating one node at a time in N’ which is nothing but the serial mode. Now, since N’ is operating in a serial mode, when N’ reaches a stable state one of the following things happen. 1. The current state of operation of both the partitions P1 and P2 that correspond to N’ may be the same which means that both P1 and P2 converge to a stable state. 2. The current state of operation of both P1 and P2 are distinct which implies N will oscillate between the two states at which P1 and P2 are existing currently, thus converging to a cycle of length two. The graphs which depict the operation in the parallel mode are shown below. Parallel mode convergence of a complex valued neural network 250 200

Energy function E

150 100 50 0 -50 -100 -150 1

2

3

4

5

6 Time t

7

8

9

10

11

Complex Valued Neural Associative Memory on the Complex Hypercube

105

Oscillation of the difference in energies between two consecutive states as the energy oscillates 250

+ +

200

Energy difference Delta E

150

100 +

+

+

50

0

-50 +

+

+

-100 + -150 1

2

3

4

5

6

7

8

9

Time t

The above graphs depict the oscillation of the value of the energy function for a peculiar case and that of the value of ∆E as it oscillates about zero.

6.4 CONCLUSIONS From the above discussion one can observe that the evaluation of the signum function at each node and thereby determining the vector that originates for the next instant makes the complex valued neural network similar in behavior to the real valued one. However the designs of neural networks( [ZURADA 1996],[ZURADA 2003] and [HIROSE 2003]) proposed so far have not seen the plausibility of the implementation of the above mentioned method of performing the complex signum function. Since the application of the function proves that the network converges just as the real valued one, it can be conveniently applied to applications such as image processing and pattern recognition.

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REFERENCES [1]. [HOPFIELD 1982] J.J. Hopfield and D.W. Tank “Neural Computations of Decisions in Optimization Problems” in Proc. Nat. Acad. Sci. USA , Vol. 79., pp. 2554-2558, 1982. [2]. [BRUCK 1987] Jehoshua Bruck and Joseph W.Goodman “A Generalized Convergence Theorem for Neural Networks”. IEEE First Conference on Neural Networks, San Diego, CA June 1987. [3]. [ZURADA 1996] Stainslaw Jankowski, Andrzej Lozowski and Jacek M.Zurada, “Complex-valued Multistate Neural Associative Memory”. IEEE Transactions on Neural Networks. Vol. 7, No 6, November 1996. [4]. [ZURADA 2003] Mehemet kerem Muezzinoglu, Student member IEEE, Cuneyt Guzelis and Zacek.M.Zurada, Fellow IEEE “A New Design Method for the Complex-valued Multistate Hopfield Associative Memory”. IEEE Transactions on Neural Networks Vol. 14. No. 4, July 2003. [5]. [HIROSE 2003] Akira Hirose. “Complex Valued Neural Networks: Theories and Applications”. World Scientific Publishing Co, November 2003. [6]. G. Rama Murthy and D. Praveen, “Complex valued Neural Associative Memory on the Complex Hypercube,” Proceedings of 2004 IEEE Conference on Cybernetics and Intelligent Systems (CIS 2004).

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CHAPTER

7

Optimal Binary Filters: Neural Networks

7.1 INTRODUCTION In the case of many artificial/natural linear/non-linear systems/channels the output signal is corrupted by noise. A natural important practical/theoretical problem is filtering. Thus it is also desirable to design an optimal filter. In many traditional approaches, the criterion of optimality is the minimization of mean square error between actual output and estimated output. Based on this criteria Wiener formulated the problem and discovered the optimal filter. This filter is derived based on transfer function description. Kalman formulated the problem based on state space description of linear system and in his case the filter is recursive. Independent of the research in system theory, Shannon formalized information theory by formulating the notion of block codes for noisy communication channels. The relationship between system theory approach and coding theory approach was seriously investigated by few researchers. In [2] the problem of optimal signal design for linear system/channels was formulated and solved. It is shown in this chapter that investigating the relationship between system theory approach and coding theory approach leads to the formulation and solution of a new optimal filtering problem. In section 2 an optimal signal design problem is formulated and solved which then forms the basis of the optimal filtering problem and its solution which is discussed in section 3. Finally the section 4 concludes the chapter.

7.2 OPTIMAL SIGNAL DESIGN PROBLEM: SOLUTION Our essential goal is to formulate and solve an optimal filtering problem. But the problem is related to optimal signal design problem, which is formulated and solved in the following. The signal design problem is an open research problem formulated by A.Wyner, in continuous time [8].

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7.2.1 Problem statement Find an admissible sequence of input signals (possibly vectors) u k , k = 0, 1, 2,..., k f –1 i.e, ith

i component of which (vector) is bounded in amplitude by one i.e. uk ≤ 1, in order to k −1

minimize the criterion J =

f −1 XkT CT CXk 2 k =0

∑

subject to

X k +1 = AX k + Buk , X (0) = 0 Yk = CX k

Here A is n × n matrix, B is n × p matrix, C is an m × n matrix. Single input, single output (SISO) as well as multi-input, multi-output (MIMO) channels are considered. Let the m × p impulse response matrix [6] be denoted by h(.).

7.2.2 Solution The optimal control vectors which maximize the total output energy of a linear discrete time filter over a finite horizon [0, k f ] are given by  u k = sign   k f − k −1

where Rkj =

∑

i = max{1, j − k −1}

j



* kj u j 

∑R



 hT (i + 1)h(k + i + 1 − j ) and u * is the optimal choice. The condition k

provided above provides a necessary condition on the optimum input signal/control. The stable states of a neural network constitute the local optimum control vector. The global optimum stable state provides the global optimum control vector [2]. Proof : Discrete Maximal Principle well known in literature is utilized to provide the solution. Consider a one-dimensional linear dynamical system. Its state space description is given by

X(k +1)= A(k) X(k) + B(k)u(k), X(0)=X 0

(7.1)

Y(k)=C(k) X(k) + D(k) u(k); Here, D(k) ≡ 0 for all' k '

(7.2)

It should be noted that we are purposefully considering a linear time varying system. In the above state space description of the linear system, A(k ) is an ‘n × n’ matrix (a second

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order tensor) and X(k ) is the state vector (first order tensor) i.e. an ‘n × 1’ vector. C(k ) is an ‘m × n’ matrix, B(k ) is an ‘n × p ’ matrix. In the case of certain multidimensional linear systems as well as infinite dimensional linear systems, the state transition tensor [7], the state tensor, the input tensor [5], the output tensor are of compatible dimension as well as order. The discrete time dynamical system evolution (linear or non-linear) is described through tensors [7]. The inner product between the linear operators is carried out with the standard method. To restrict oneself to the problem considered in the present chapter, we return to the one-dimensional system keeping in mind that the authors already made the extension to multi/infinite dimensional systems [7]. i The input sequence satisfies the constraints of the following form i.e. uk ≤ 1 , where u ki

is the ith component of the input vector. Thus, U k ∈ V , a subset of Rr. The cost function J is given below. The problem considered is to find an admissible sequence uˆ k , k = 0,1,..., k f − 1 subject to the constraints and also minimizing the objective function J. k −1

f 1 T T −1  X kT CT (k )C(k )X k  − Xk f C ( k f )C( k f )Xk f J= 2 2 k =0

∑

(7.3)

Let uk , k = 0,1,..., k f − 1 be an optimal sequence, and let xk , k = 0,1,..., k f be the state response of (7.1) uniquely specified by u. Then, under reasonable assumptions, there exists a non-trivial function satisfying

λk =

∂H( xk , uk , λk , k ) ∂xk

(7.4)

where, the Hamiltonian is given by H (x k , uk , λk +1 , k ) =

−1 T T  Xk C ( k )C( k )Xk  +λ T [ A( k )X + B(k )u ] k +1 k k 2 

(7.5)

as in most textbooks [9]. Thus, the adjoint vector equation is given by

λk = −CT (k )C(k )Xk + AT (k )λk +1

(7.6)

Since, the terminal state is unspecified, we have

λ (k f ) = −C T (k f )C( k f )X (k f )

(7.7)

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This will provide the terminal condition for solving (7.6). Since the input is constrained it must necessarily satisfy

( H (xk , v , λk +1 , k )) H (xˆ k , uˆ k , λˆk +1 , k ) = min v∈V

(7.8)

for all k = 0,1, 2, …. k f −1 Thus,

T uk* = − S ign {B ( k )λk + 1 }

(7.9)

In most textbooks [6] and references for optimal control /signal design (7.9) is derived as a necessary condition. We make the following detailed derivation, to expose the structure of optimal control and its relationship to stable states of a neural network. Solving (7.6) for λ k +1 and substituting in (7.9) we arrive at the optimal control sequence. It is immediate to see that if v is a convex polytope, then we have a mathematical programming problem. Our chief contribution is the following derivation. With the terminal state specified, the equation (7.6) is recursed backwards to arrive at the optimal vector (optimal control tensor in the multi–dimensional case). Thus, an efficient computational form for solving the two-point boundary value problem [9] is derived in the following. It should be noted that, we derived the expression for λ k+1 .in the case of linear time varying dynamical system.

λk = −CT (k)Y(k ) + AT (k )λk +1 with λ (k ) = −C T ( k )Y( k ) f f f Starting with the terminal condition, recursing backwards λk f −1 = −C T ( k f − 1)C( k f − 1)X ( k f − 1) − AT ( k f − 1)CT ( k f )C( k f )X( k f ) T = −C ( k f − 1)Y (k f − 1)

− AT ( k f − 1)CT ( k f )Y( k f )

λk f −2 = −C T ( k f − 2)C( k f − 2)X (k f − 2) − AT (k f − 2)C T (k f − 1)Y (k f − 1) − AT ( k f − 2) AT ( k f − 1)C T ( k f )Y( k f )

λk f −3 = −CT ( k f − 3)Y( k f − 3)

(7.10)

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− AT ( k f − 3)CT ( k f − 2)Y( k f − 2) − AT ( k f − 3) AT ( k f − 2)C T ( k f − 1)Y( k f − 1) − AT ( k f − 3) AT ( k f − 2) AT ( k f − 1)CT ( k f )Y( k f )

Thus continuing the pattern downwards, we have for the linear time invariant filters

λk f −l = −CT Y( k f − l) − AT CT Y( k f − l + 1) − ( AT )2 C T Y ( k f − l + 2) − ( AT )3 C T Y ( k f − l + 3) ... −( AT )l C T Y ( k f )

(7.11)

Let l = k f − k − 1; this implies k f − l = k + 1 Hence

λk +1 = −CT Y(k + 1) − AT CT Y(k + 2) −( AT )2 CT Y(k + 3) ... −( AT )

k f − k+l

(7.12)

C T Y( k + l + 1)

Thus, we have for the linear time invariant filters *  l  uk = Sign  BT ( AT )i C T Y ( k + i + 1)   i =0 

∑

(7.13)

where, l = k f − k − 1 . Using the expression for impulse response h. *  l  i.e. uk = Sign  h T (i + 1)Y ( k + i + 1)   i =0 

∑

(7.14)

The term in the parentheses is given by, l

∑h i =0

T

(i + 1)Y( k + i + 1) =

l

∑ i =0

k + i +1

hT (i + 1)

∑ h (k + i + 1 − j)u ( j) j =0

(7.15)

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Exchanging the order of summation, with the help of grid, in Figure 7.1 we have

uk*

 k f = Sign   j =0 i

∑

 [h T (i + 1) h (k + i + 1 − j )] u ( j ) j − k −1) 

k f −k −1

∑

= max ( 1,

kf

k+1

0, 0

1

kf – k – 1

2

Fig. 7.1 Order of summation

Writing the expression

uk*

k f − k −1  k f  [h T (i + 1)h (k + i + 1 − j )]u ( j ) Sign =   j =0 i =max{1, j − k −1} 

∑

∑

(7.16)

Let us define k f − k −1

Rkj =

∑

i = max{1, j −k −1}

 hT (i + 1) h ( k + i + 1 − j )

(7.17)

Thus,  k f  uk* = Sign ∑ Rk ju ( j )  j =0 

(7.18)

In the case of linear time varying systems, the above derivation still applies, as is easily seen above. It is easy to see that we have an expression of the following form for the optimal control vector over finite horizon of a time varying linear system.

uk* = Sign

{∑ S

}

kj u( j )

where, Skj is the energy density matrix of time varying linear system. This can be stated as a theorem. The authors derived similar results for multidimensional and infinite dimensional systems[2].

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7.3 OPTIMAL FILTER DESIGN PROBLEM: SOLUTION (DUAL OF SIGNAL DESIGN PROBLEM) The optimal filter problem is formulated below. For simplicity we consider a single input, single output linear filter/channel/system. It is easily seen that extension to multi input, multi output systems follows in a straightforward manner. It is discussed how the solution of optimal signal design problem also leads to a solution to this problem.

7.3.1 Problem statement Find an admissible sequence of impulse response values h k , k = 0, 1, 2,..., k f –1, which are

bounded in amplitude by one i.e. hk ≤ 1, in order to minimize the criterion.

−1 J = 2

k f −1

∑X

T T k C CX k

X k +1 = AX k + Buk ,

subject to

X (0) = 0

Yk = CX k where A is n x n matrix , B is n x 1 matrix, C is an 1 × n matrix. Let the impulse response at time ‘k’ be denoted by h(k ). That is, find a bounded support ,bounded magnitude impulse response values such that total output energy over a finite horizon is maximized. The input is unconstrained.

7.3.2 Solution Since convolution is a commutative operator, as far as the output of linear filter is concerned, the roles of input and impulse response can be exchanged. y ( n ) = u ( n ) * h ( n) = h ( n ) * u ( n )

The input and impulse response have a dual role in determining output. Maximizing output subject to bounded extent, bounded support input is equivalent to maximizing output subject to bounded extent, bounded support impulse response. The optimal input vector is given as the stable state of a neural network [3]. Thus optimal input signals constitute a linear code [1], [4]. The optimal set of impulse responses constitutes stable states of a neural network whose connection matrix is the input energy density matrix. The components of optimal impulse response vector assume binary values. They constitute a linear code. The linear filtering operation reduces to the “binary filtering” operation. The optimal binary filters are related to optimal codes matched to input. The derivation follows by replacing the “input” by “impulse response” finite in extent and finite in support. The derivation involves duplication

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effort required with the derivation of “optimal input”. Thus, linear filtering involves weighting the input values in the window by binary values. It is shown in [3] that the logic functions in multidimensions also constitute the stable states of an m-d neural network. Known impulse response of linear channel over finite horizon

Determine the connection matrix of neural network (energy density matrix)

Determine the local/global optimum input signal, i.e., the stable state of neural network by running it in serial mode

Fig. 7.2 Optimal signal design approach

Known input signal over a finite horizon

Construct the input energy density matrix, the connection matrix of a neural network

Run the above neural network in serial mode to compute the local/global optimum impulse response

Fig. 7.3 Optimal filter design approach

7.4 CONCLUSIONS In the above discussion, a realistic/ practical optimal filtering problem is formulated. It is shown that this filtering problem is related to an optimal control problem [6]. By solving

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the optimal control/ signal design problem, it is shown that the global optimum impulse response constitutes the stable state of a Hopfield neural network.

REFERENCES * Journal: [1] Jehoshua Bruck, Mario Blaum, “Neural Networks, Error – Correcting Codes, and Polynomials over the Binary n- Cube“, IEEE Transactions on Information Theory, Vol. 35, No. 5, September 1989. * Conference Proceedings: [2] G. Rama Murthy, “Optimal Control, Codeword, Logic Function Tensors: Multidimensional Neural Networks,” International Journal of Systemics, Cybernetics and Informatics (IJSCI), October 2006, pages 9-17. [3] G. Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic theory,” International Journal of Neural Systems, Vol. 15, No. 3, pp. 223-235, 2005. [4] G. Rama Murthy, “Multi/Infinite Dimensional Coding Theory: Multi/Infinite Dimensional Neural Networks: Constrained Static Optimization,” Proceedings of IEEE Information Theory Workshop, October 2002. * Books: [5] A.I. Borisenko and I. E. Tarapov, “Vector and Tensor Analysis with Applications“, Dover Publications Inc., New York, 1968. [6] M. Gopal, “Modern Control System Theory“, John Wiley and Sons, New York. [7] G. Rama Murthy, “Tensor State Space Representation: Multidimensional Systems”, International Journal of Systemics, Cybernetics and Informatics (IJSCI), January 2007, pp.16-23 [8] B. Gopinath, T.Cover , “Open Problems in Control and Communication and Computation.”, Springer, Hiedelberg, 1987. [9] A.E. Bryson and Y.C. Ho, “Applied Optimal Control: Optimization, Estimation and Control”, Taylor and Francis Inc. 1995.

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CHAPTER

8

Linear Filter Model of a Synapse: Associated Novel Real/Comple xV alued Neural eal/Complex Valued Networks

8.1 INTRODUCTION Artificial neural networks are innovated to provide models of biological neural networks. The currently available models of neurons are utilized to build single layer ( e.g. single layer perceptron ) as well as multi-layer neural networks (e.g. multi-layer perceptron). These neural networks were utilized successfully in several applications. Also various paradigms of neural networks such as radial basis functions, self-organizing memory are innovated and utilized in applications. In the case of conventional real valued neural networks, the inputs, outputs belong to the Euclidean space (Rn or Rm ). In these neural networks, a synapse is represented/ modeled by a single synaptic weight which is lumped at one point. These synaptic weights are updated in the training phase using one of the learning laws ( for example, Perceptron learning law, gradient rule etc). In the case of supervised training, these learning laws enable one to classify the input patterns into finitely many classes (based on the training samples). Motivation for a Better Model of Neurons: • By reflecting on modeling biological neurons, we are naturally led to making the realistic assumption that synapses constitute distributed elements rather than lumped elements. Thus, a realistic model of a synapse is a linear system (characterized by impulse response) while at the same time maintaining tractability. • In conventional neuronal models, the input at each synapse is a constant and is acted on by the scalar synaptic weight. But in biological neurons, it is most natural to consider that the input signal samples are not scalar values, but are functions defined over a finite support. The synapses (characterized by impulse response) act on these input signals which are defined on the domain (restricted to a

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support) [0,T]. Thus the class of input signals belong to a function space (defined on [0,T]). For the sake of notational convenience, let the synaptic weight functions be also defined on [0,T]. In summary, a continuous-time, real valued neuron has input signals (which are real valued functions of time) defined over a finite support. The input signals are fed to synapses acting as linear systems/filters and sum of responses is operated on by an activation function. Using this model of a neuron, various feed-forward/recurrent networks of neurons are designed and studied. This chapter is organized as follows. In section 2, continuous time perceptron model is discussed. Also in this section, the continuous time perceptron learning law is discussed. In section 3, abstract mathematical structure of neuronal models is discussed. In section 4, neuronal model based on finite impulse response model of synapse is discussed. Also the associated neural networks are proposed. In section 5, a novel continuous time associative memory is proposed and the convergence theorem is discussed. In section 6, various multidimensional neural network generalizations are discussed. In section 7, complex valued neural networks based on the continuous time neuronal model are discussed. The chapter concludes in section 8.

8.2 CONTINUOUS TIME PERCEPTRON AND GENERALIZATIONS The area of artificial neural networks was pioneered by the efforts of McCulloch and Pitts to provide a model of neuron. Soon, it was realized by Minsky et al. that such a model of neuron has no training of the synaptic weights. Thus they proposed the model of single perceptron as well as single layer of perceptrons. Further they provided the perceptron learning law. This law was proved to converge when the input patterns are linearly separable. Later it was shown that a Multi-Layer-Perceptron, a feed forward network can be trained (using the back-propagation algorithm) to classify non-linearly separable patterns. In the following (as discussed in the Introduction), we propose a more accurate (biologically) model of neuron and use it to construct various artificial neural networks.

A New Mathematical Model of Neuron/Single Perceptron Consider finitely many (say M) input signals which are defined on a bounded support [0, T]. Let each of these signals be input to synapses which are characterized by synaptic weight functions (that are defined on support [0, T] ). Since each of the synapses act as a linear filter, the output of each synapse is a convolution of the input function with the synaptic weight function. Mathematically, let a i(t), Wi(t) for 1≤ i ≤ M be the input functions , synaptic weight functions respectively. Let the signum function be the activation function of the neuron. Thus the output of the neuron is given by

M  y(t) = Sign  ai (t ) ⊗ Wi (t ) − T   i =1 

∑

(8.1)

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119

where ⊗ denotes the convolution operation between two time functions (and T is the threshold at the neuron. Without loss of generality, T can be assumed to be zero ). More explicitly,

 M y(t) = Sign      i =1

T

∑∫ 0

  ai (τ ) Wi (t − τ ) dτ  − T     

(8.2)

The successive input functions are defined over the interval [0,T]. They are fed as inputs to the continuous time neurons at successive SLOTS. M  y(t) = sign  ∑ ai (t ) ⊗ w i (t )  i =1 

a1 (t)

w1 (t) w2(t)

a2 (t) wm(t)

. . . am (t)

Fig. 8.1 A novel model of continuous time neuron (In the figure ⊗ denotes the convolution operator)

Continuous Time Perceptron Learning Law: Proof: As in the case of “conventional perceptron”, a continuous time perceptron learning law is given by:

Wi( n+1) (t) = Wi(n ) (t) + η ( S(t ) − g(t) ) ai (t)

(8.3)

where S(t) is the target output for the current training example, g(t) is the output generated by the perceptron andη is a positive constant called the learning rate. Proof: In this model of continuous time perceptron, the weights are functions of time defined on the interval [0, T]. Thus, since the synaptic weights are functions of time, we are led to investigating the type of convergence: (i) Pointwise or (ii) Uniform. Suppose we fix the time point, t. The convergence of synaptic weights in (8.3) is assured by the proof of convergence in the case of conventional perceptron. Since, the choice of time point is arbitrary, we are assured of pointwise convergence of synaptic weights based on training sample input functions. It is interesting to know under what conditions, the sequence of synaptic weight functions converge UNIFORMLY. Q.E.D.

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Continuous Time Multi-Layer-Perceptron: Using the above approach to model a neuron, it is straightforward to arrive at a multilayer feed forward network. In such a multi-layer perceptron, the activation function at each neuron is changed from being a signum function to a sigmoid function i.e. y(t) = where y(t) is the output of neuron,

1 , 1 + e − z(t )

(8.4)

M

z(t) =

∑ a (t) ⊗ W (t) i

j =1

i

( ⊗ ..convolution operator)

(8.5)

The generalization of back propagation algorithm (based on conventional model of neuron) follows essentially in a one-to-one manner. The details are avoided for brevity. Also various recurrent networks based on the continuous-time neuron can be designed and implemented. It is possible to consider a model of neuron in which the input functions are defined over the function space [0, ∞ ]. It is possible to consider neural networks based on such inputs. The inputs are divided into testing and training classes.

8.3 ABSTRACT MATHEMATICAL STRUCTURE OF NEURONAL MODELS Consider the inputs to a continuous-time neuron which are defined on a finite support [0,T]. Let the impulse responses of synapses modeled as linear filters be defined on the finite support [0,T]. Thus, the inputs as well as synaptic weight functions belong to the function space defined over the finite support [0,T]. We answer the following question. Q: Under reasonable assumptions, what is the mathematical structure of the function space defined over [0,T] ? Let F be the set (function space) on which the following operations are well defined: Addition, Convolution (These operations are like addition, multiplication defined on the sets: real numbers, complex numbers). Lemma: Let the identically zero function be the additive identity element and Delta function (δ (t ) = 1for t = 0 and δ (t ) = 0 for t ≠ 0 ) be the multiplicative identity. Then, the set F on which addition, multiplication (of functions defined on [0,T]) operations are defined constitutes a Ring. Proof: Involves routine verification of axioms of the ring (closure under addition, convolution operations between the members of F i.e. functions) and are avoided for brevity. Actually F is “Close” to being a field except that “multiplicative inverse” does not always exist. Q.E.D. Include such functions and convert F into field. Now define a vector space, G over the field. The set of input functions incorporated into a vector belongs to G. The usual ‘multiplication’ operation is replaced by ‘convolution’.

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• Hyperplane: In the vector space defined above, a ‘hyperplane’ defined by a ‘vector’ (specified by synaptic weight functions { Wi (t ), 1 ≤ i ≤ M } is given by M

∑ a (t) i =1

i

⊗ Wi (t) = L(t)

(8.5)

• Linear Separability: Consider the ‘field’, F of functions defined over [0,T].Let G be the vector space defined over F. A class of functions is separable into two classes, if there exists a hyperplane such that the two regions are defined by M

∑ i =1

ai (t) ⊗ Wi (t ) ≤ L(t ) and

M

∑ a (t) i =1

i

⊗ Wi ( t) > L( t)

(8.6)

Similarly, it is straightforward to define the class of functions which are classifiable into “N” classes.

Fourier/LAPLACE Transform: Associated Field It is well known that the Fourier/Laplace transform of the convolution of two functions, is the product of Fourier/Laplace transforms of the individual functions. It is found that processing the functions (by applying the activation function) has advantages in the transform domain (Rama 5). In this subsection, we restrict attention to functions with rational Fourier/Laplace transforms. The function space being operated on by the activation function is now the field of rational functions over [0,T]. Thus there is a natual mapping between the two fields (associated with continuous time neuron). With the above discussion summarizing the abstract mathematical structure of neuronal modeling (being considered), we arrive at the following conclusions: • Consider a single layer of continuous time perceptrons being trained by input function samples. As long as the input samples are “linearly separable”, the set of synaptic weight functions converge (to an equilibrium vector) • Consider a continuous-time multi-layer perceptron being trained by input function samples. The back propagation algorithm utilized to train synaptic weight functions converges even when the input function samples are nonlinearly separable (provided there are sufficient number of continuous-time neurons in the hidden layer).

8.4 FINITE IMPULSE RESPONSE MODEL OF SYNAPSES: NEURAL NETWORKS • So far we have considered continuous time neural networks in which the synaptic weight function corresponds to an analog linear filter. A natural question arises whether it is possible to conceive a synapse whose impulse response corresponds to that

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of a digital filter i.e. a Finite Impulse Response Filter (FIR). In the following, we consider neural network with such a model of synapse. • Typically, let the discrete time input signals be considered over the finite horizon [ 0, 1, 2, …, S]. For the sake of simplicity, let the length of all FIR filters modeling the synapses be the same, say T (The generalization to the case where the FIR filters have different lengths is straightforward). Thus the impulse response sequences (associated with different synapses) extend over the duration {0,1,2,……….,}. • The output of the synapse (described by an FIR filter) depends on the input signal values over a finite horizon (depending on the length of the impulse response). Typically the length of filter is smaller than the support of a distinct input sequence i.e. T << S. It should be noted that the successive input sequences are of same length.

 M i  Sign C ( n) ⊗ a i ( n)   y (n ) =  i =1 

∑

(8.7)

M T  C i (k ) a i (n − k )  = Sign   i =1 k = 0  Where C i(k) for k = 1, 2,...,T is the impulse response sequence of ith synapse and

∑∑

a i (k ),... for k = 1, 2,..., S is the ith input sequence to the neuron. • Thus the synaptic weight sequence values (impulse response of FIR filters) can be trained according to the following perceptron learning law ( n) Ci(n+1) (k ) = Ci ( k ) + η ( S( k ) − g( k )) a i(k )

(8.8)

where S(k ) is the target output for the current training example, g(k ) is the output generated by the perceptron at time k and η is a positive constant called the learning rate. This update rule converges when the input patterns are linearly separable. Using the same model of neuron, a multi layer perceptron is trained using a modified version of Back Propagation Algorithm. It is possible to consider neuronal models in which the synapse acts as an Infinite Impulse Response filter. Furthermore, based on such a model of neuron (synapse acting as an FIR filter), it is possible to discuss a novel associative memory. Currently, the models of neurons discussed (in section 2, section 4) are being compared with traditional models of neurons [Rama5].

8.5 NOVEL CONTINUOUS TIME ASSOCIATIVE MEMORY In addressing, the problem of signal design for magnetic/optical recording channels, Wyner formulated an open research problem [GoC]. The problem statement is provided below.

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Open Research Problem Consider a single input, single output linear time invariant filter (SISO linear filter) modeling a magnetic/optical recording channel. Let the class of inputs (to the linear filter) defined on bounded support [0, T], be bounded in magnitude by unity (1). Determine the optimal signals such that the total output over finite horizon is maximized. T

∫ y (t) 2

dt ( where y(t) is the output of linear filter )

0

The author [Rama3] as well as Honig et al. [HoS] independently solved the problem. The solution in [Rama3] is more general in the sense that we considered Multi-Input, MultiOutput (MIMO) linear time varying filters and derived the optimal input vector. Let Y (t) be an optimal input vector. Then it satisfies the following signed integral equation

T  Y (t) = Sign  R (t , u) Y(u) du  0 

∫

(8.9)

where R (t,u) is the energy density matrix of the multi-input, multi-output, Linear time varying system. In the case of multi-input, multi-output, linear time invariant system, the optimal input vector satisfies the following equation

T  (t) = Sign  R (t − u) Y (u) du  Y   0 

∫

(8.10)

In the following paragraph, we consider a successive approximation procedure for computing the optimal control vector starting with an arbitrary binary vector defined on the support [0,T]. Consider a vector of binary valued functions Y ( n) (t) ( +1 or − 1valued) defined on the finite support [0,T]. Let R(t) be the energy density matrix of a multi-input, multi-output linear system representing the time varying synaptic weight matrix. The following successive approximation scheme is used to compute the local optimum stable function starting with a initial binary vector Y(0) (t).

T  (n) (8.11) (t) = Sign  R ( t − τ ) Y (τ ) d τ  • Y  0  From practical considerations, it is necessary to know whether the above successive approximation scheme converges or not. This problem is converted into an equivalent problem by discretizing the continuous time linear system into a discrete time system. Such discretization can always be done for some types of systems (satisfying some regularity ( n +1)

∫

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conditions) without fear of approximating the system dynamics. The standard procedure of discretizing a continuous time system is summarized in many textbooks including Gopal’s book ( Gop., Pages 185-187), With the discrete time system equivalent to the continuous time system, the argument technique adopted for convergence is once again the energy function hill climbing in successive iterations. Theorem 8.1: Consider a Multi-Input, Multi-Output (MIMO), linear time-invariant system described by the dynamics i X (t ) = A X (t ) + B Y (t ) Z(t) = C X (t)

(8.12) The discrete time simulation (of the above continuous time system) of the following form X( k+1 ) = F X( k ) + G Y (k)

(8.13)

Z( k ) = H X( k )

(8.14)

can always be done. The discrete simulation is almost exact except for the error introduced by sampling the input and that caused by the iterative procedure for evaluating the matrices. Proof: Follows from the procedure described in Gopal (Gop, pp.185-187 ). Q.E.D. With such a discrete time system corresponding to a continuous time system, we have the following recursion (successive approximation scheme):

Y (n +1) ( k ) = Sign {W Y( n) (k )}

for n ≥ 0,

(8.15)

Where Y(k ) is the optimal control vector associated with the discrete time linear system (obtained by discretizing a continuous time system) and W is the energy density tensor (associated with the discrete time system). Thus we have a Hopfield network with W as the synaptic weight matrix. Hence starting with an initial vector Y (0) (k ) , the above recursion converges to a stable state (local optimum vector) or at most a cycle of length 2 ( by invoking the convergence theorem associated with Hopfield neural network whose Connection matrix is W). Q.E.D. Thus, the above approach converts the problem of determining the convergence of scheme in (8.11), to that associated with a discrete time linear system. The iteration reminds of L∞ version of Neumann Series. The energy function (Lyapunov function) optimized over the state trajectory of continuous time linear system is a quadratic form [Rama1]. In [BrB], various possible generalized neural networks are discussed. These neural networks are associated with an energy function which is a higher order form than a quadratic form (associated with a Hopfield neural network). It is very natural to formalize associative memories which are generalizations of those discussed in this chapter. Several generalizations of the results are documented in the technical report [Rama5]. For instance, the complex valued, continuous time associative memory is discussed in

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detail in the technical report [Rama5, RaP]. For such a complex valued associative memory, a convergence theorem is stated and proved.

8.6 MULTIDIMENSIONAL GENERALIZATIONS • In this chapter, so far, we have considered single/multi-layer continuous time neural networks, whose input as well as output are vectors. It is straight forward to generalize the results to the case where the input/output is a 3dimensional/multidimensional array [Rama1, Rama 2]. Tensor products are utilized to determine the output of each neuron in the network. Such three/ multidimensional neural networks arise in the biological neural network in human/animal brain. • In the case of a human/animal brain, the associative memory operates on three dimensional input patterns. Thus the state of the associative memory is not a vector (one dimensional array) but a three dimensional array. An appropriate model of such memory is a three dimensional, continuous time associative memory. It is easy to see that the model described in section 5 can easily be generalized along the lines of the work in [Rama 2].

8.7 GENERALIZATION TO COMPLEX VALUED NEURAL NETWORKS (CVNNs) Activation Functions Consider a complex valued, continuous time neuron whose inputs as well as synaptic weight functions (defined on support [0, T] ) and thresholds are complex valued functions. In such a model of neuron, it is possible to utilize various activation functions. Let z(t) = ( c(t) + j d (t) ) be the net contribution (after convolving the input functions with the synaptic weight functions) at a neuron. The following activation functions can be utilized: Activation Functions: (i) Complex Signum Function; Sign ( c(t) + j d (t) ) = Sign ( c(t) ) + j Sign ( d (t) ). With such an activation function, the continuous time perceptron convergence law described in (8.3) for real values neurons is easily generalized to continuous time, complex valued perceptrons. In the case of conventional complex valued perceptron, it is well known that the perceptron training law is easily generalized [AAV]. Using the similar proof technique, in the case of complex valued, continuous time neurons, the convergence proof utilized by Aizenberg et al. is generalized. Also, in the case of conventional, complex valued neuron, the above activation function is utilized in [RaP] for arriving at an associative memory.

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(ii) Complex Sigmoid Function: 1 g(z(t)) = 1 + e − z(t ) or alternatively

(8.16)

g(z(t)) = tanh ( z(t))

(8.17)

In the case of complex valued, continuous time multi-layer perceptron, we utilize the above complex valued sigmoidal function as the activation function at each (complex valued) neuron. With such a model of neuron, the backpropagation algorithm in Nitta et al. [Nit1, Nit2] is generalized to the case of continuous time neural networks. Utilizing traditional model of a neuron, unified theory of control, communication and computation is discovered and formalized [Rama 4]. This unified theory is generalized using the models of neurons discussed in this chapter [Rama1].

8.8. CONCLUSIONS In this, chapter models of neurons are proposed. The synapses are considered as distributed elements rather than lumped elements. Thus, synapses are modeled as linear filters in continuous time as well as discrete time. Using these novel models of neurons, associated neural networks are proposed. Also, a novel model of associative memory is proposed. Using such a model, convergence aspects of various modes of operation is discussed. Multidimensional generalizations of neural networks are discussed. Also associated complex valued neural networks are discussed.

REFERENCES [AAV] I. N. Aizenberg, N. N. Aizenberg and J. Vandewalle, “Multi-Valued and Universal Binary Neurons”, Kluwer Academic Publishers, 2000. [BrB] J. Bruck and M. Blaum, “Neural Networks, Error Correcting Codes, and Polynomials over the Binary n-Cube”, IEEE Transactions on Information Theory, pp. 976- 987, Vol. 35, No.5, September 1989. [GoC] B. Gopinath and T. Cover, “Open Problems in Control, Communication and Computation”, Springer, Hiedelberg, 1987. [Gop] M. Gopal, “Modern Control System Theory“, John Wiley & Sons, Second Edition, 1993. [HoS] M. Honig and K. Stieglitz, “On Wyner’s Conjecture”, Bellcore Technical Memorandum. [Nit1] T. Nitta and T. Furuya: “A Complex Back-propagation Learning”, Transactions of Information Processing Society of Japan, Vol.32, No.10, pp.1319-1329 (1991) (in Japanese).

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[Nit2] T. Nitta : “An Extension of the Back-Propagation Algorithm to Complex Numbers”, Neural Networks, Vol.10, No.8, pp.1391-1415 (1997). [Rama1] G. Rama Murthy,” Unified Theory of Control, Communication and Computation”, To be submitted to Proceedings of IEEE. [Rama2] G. Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory“, International Journal of Neural Systems, Vol. 15, No. 3, Pages 223-235, June 2005. [Rama3] G. Rama Murthy, “Signal Design for Magnetic/Optical Recording Channels: Spectra of Bounded Functions“, Bellcore (Now Telcordia) Technical Memorandum, TM-NWT-018026. [Rama4] G. Rama Murthy, “Optimal Control, Codeword, Logic Function Tensors: Multidimensional Neural Networks“, International Journal of Systemics, Cybernetics and Informatics, October 2006, pages 9-17. [Rama5] G. Rama Murthy, “Linear Filter Model of Synapses: Associated Novel Real/Complex Valued Neural Networks“, IIIT Technical Report in Preparation. [RaP]G. Rama Murthy and D. Praveen, “Complex-Valued Neural Associative Memory on the Complex Hypercube“, Proceedings of 2004 IEEE International Conference on Cybernetics and Intelligent Systems (CIS-2004), Singapore.

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CHAPTER

9

Novel Comple xV alued Complex Valued Neural Networks

9.1 INTRODUCTION Starting in 1950s researchers tried to arrive at models of neuronal circuitry. Thus the research field of artificial neural networks took birth. The so-called, perceptron was shown to be able to classify linear separable patterns. Since the Ex-clusive OR gate cannot be synthesized through any perceptron (as the gate outputs are not linearly separable), the interest in artificial neural networks faded away. In the 1970s, it was shown that multi-layer feed forward neural network such as a multi-layer perceptron is able to classify non-linearly separable patterns. Living systems/machines such as homosapiens, lions, tigers etc. have the ability to associate externally presented one/two/three dimensional information such as audio signal/images/three dimensional scenes with the information stored in the brain. This highly accurate ability of association of information is amazingly achieved through the bio-chemical circuitry in the brain. In 1980s Hopfield revived the interest in the area of artificial neural networks through a model of associative memory. The main contribution is a convergence theorem which shows that the artificial neural network reaches a memory/ stable state starting in any arbitrary initial input (in a certain important mode of operation). He also demonstrated several interesting variations of associative memory. In (Rama4), a continuous-time version of associative memory is described. It is shown that the celebrated convergence theorem in discrete time generalizes to the continuous time associative memory. In (Rama2), the model of associative memory in one dimension (Hopfield associative memory) is generalized to multi/infinite dimensions and the associated convergence theorem is proven. It was realized by researchers such as N.N. Aizenberg that the basic model of a neuron must be modified to account for complex valued inputs, complex valued

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synaptic weights and thresholds [AAV]. In many real world applications, complex valued input signals need to be processed by neural networks with complex synaptic weights [Hir]. Thus the need to study, design and analysis of such networks is real. Also, in (Rama3) the results on real valued associative memories are extended to complex valued neural networks. In [Nit1, Nit2], the celebrated back propagation algorithm is generalized to complex valued neural networks. Also, in [Rama4], based on a novel model of neuron, complex valued neural networks are designed. Thus, based on the results in section 2, section 3, it is reasoned that transforming real valued signals into complex domain and processing them in the complex domain could have many advantages. This chapter is organized as follows. In Section 2, Discrete Fourier Transform (DFT) is utilized to transform a set of real/complex valued sequences into the complex valued ( frequency) domain. It is reasoned that, in a well defined sense, processing the signals using complex valued neural networks is equivalent to processing them in real domain. In Section 3, a novel model of continuous time neuron is discussed. The associated neural networks (based on the novel model of neuron) are briefly outlined. In Section 4, some important generalizations are discussed. In Section 5, some open questions are outlined. The chapter concludes in Section 6.

9.2 DISCRETE FOURIER TRANSFORM: SOME COMPLEX VALUED NEURAL NETWORKS In the field of Digital Signal Processing (DSP), discrete sequences are processed by discrete time circuits such as digital filters. One transform which converts the time domain information into frequency domain is called as the Discrete Fourier Transform (DFT). One of the main reasons for utilizing the DFT in many applications is the existence of a fast algorithm to compute DFT. This fast algorithm is called as the Fast Fourier Transform (FFT). In the following, we provide the mathematical expressions for the Discrete Fourier Transform (DFT) as well as Inverse Discrete Fourier Transform (IDFT) of a discrete sequence {X n } nM=−01 i.e. { x0 , x1 , x2 ,..., xM − 1 } . M −1

DFT: X ( k ) =

∑

IDFT: x(n ) =

1 M

n =0

x(n) WMk n for 0 ≤ k ≤ ( M − 1) M −1

∑ k =0

X( k ) WM− k n for 0 ≤ n ≤ ( M − 1)

(9.1) (9.2)

Where WM = e

− j(

2π ) M

(9.3)

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The results in this section are motivated by the question: Main Question: Consider a set of samples which are linearly separable in the Mdimensional Euclidean space. Utilizing an invertible (Bijection) Linear Transformation, transform the points. In the transformed domain, are the resulting samples, linearly separable? In answering this question, we are led to the following Lemma. Lemma 9.1: Under Bijective Linear Transformation, linearly separable patterns in Euclidean Space are mapped to linearly separable patterns in the transform space. Proof: For the sake of notational convenience, we consider the patterns in a 2-dimensional Euclidean space. Let the bijective/invertible linear transformation be T: R 2 → R 2 . Let the original separating line (more generally hyperplane) be given by W1 X + W2 Y = C

(9.4)

Two regions (decided by the separating line/hyper plane be) in R 2 are: S1 = {( x, y)| W1 x + W2 y ≥ C }

S2 = {( x , y)| W1 x + W2 y < C }

(9.5)

Now let us consider the Linear Transformation, T: T : R2 → R 2 ( x , y ) → ( px + qy , rs + sy )

(9.6)

Let the linear transformation be represented by the following matrix:

 p q  r s  

(9.7)

Under this transformation, the separating line coordinates become:

 X '   p q  X   Y '  =  r s  Y      

(9.8)

Thus we readily have

X‘ = pX + qY Y‘ = rX + sY On inverting the linear transformation, we have

(9.9)

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132

X  p Y =    r

q s 

−1

 X ' Y '     s /d =  −r /d

(9.10)

−q /d   X '  p /d   Y '  '

Where d is the determinant of the matrix and is given by d = p s – q r. We thus have

q s  X d X' − d Y '   Y =  p    −r X' + Y '   d d 

(9.11)

Thus, substituting for X, Y in the original separating line/hyper plane W1 X + W2 Y = C , we readily have

q p s   −r  W1  X ' − Y '  + W2  X' + Y ' = C d d d   d  (W1 s − W2 r ) X‘ + (−W1q + W2 p)Y‘ = Cd

(9.12)

From the above equations, it is clear that a point in two dimensional Euclidean space belonging to the set gets transformed to the point T (x . y ) = (x ', y ') i.e.

(x , y ) ∈ S1 T (x , y ) = ( x ', y ')∈S '1

Where the set S’1 is given by

S1‘ = {( x‘, y‘):(W1 s − W2 r )x‘ + (−W1q + W2 p)y‘ ≥ Cd}

(9.13)

• Thus we have shown that the patterns which are linearly seperable in two dimensional Euclidean space will remain linearly seperable after applying a bijective linear transformation to the samples. • The above proof is easily generalized to samples in n-dimensional Euclidean space ( where ‘n’ is arbitrary). Q.E.D. Consider the equation (9.1) for computing the Discrete Fourier Transformation of a discrete sequence of samples {x(n) : 0 ≤ n ≤ ( M − 1)}. Let the column vector containing these samples be given by Y. Also, let the column vector containing the transformed samples i.e {X(k) : 0 ≤ k ≤ ( M − 1)} be given by Z. It is clear that equation (9.1) is equivalent to the following:

Novel Complex Valued Neural Networks

Z = F Y,

133

(9.14)

Where F is the Discrete Fourier Transform matrix. This matrix is invertible. Hence the transformation between the discrete sequence vectors Y, Z is bijective. Thus the above Lemma applies.

9.3 COMPLEX VALUED PERCEPTRON Consider a single layer of conventional perceptrons. Let the sequence of input vectors be

{Y1 , Y2 ,..., YL } . The following supervised learning procedure is utilized to classify the patterns: • Apply the DFT to the successive input training sample vectors resulting in the vectors. {Z1 , Z2 ,...., ZL } . • Train a single layer of Complex Valued Perceptrons using the transformed sample vectors (complex valued version of perceptron learning law provided in [AAV] is used) • Apply the IDFT to arrive at the proper class of training samples. • Utilize the trained complex valued neural network to classify the test patterns. In view of Lemma 1, the above procedure converges when the training samples are linearly separable. Thus the linearly separable test patterns are properly classified. The above procedure is also applied for non-linearly separable patterns using a complex valued Multi-Layer Perceptron. Back propagation algorithm discussed in [Nit1, Nit2] is utilized. Detailed discussion is provided in [Rama1]. It is argued by Nitta et al. that the complex valued version of back propagation algorithm converges faster than the real one. Thus from computational viewpoint, the above procedure is attractive.

9.4 NOVEL MODEL OF A NEURON: ASSOCIATED NEURAL NETWORKS In conventional model of neuron, weighted contribution (weights being the synaptic weights) of current input values is taken and a suitable activation function ( Signum or Sigmoid or hyperbolic tangent) is applied. A biologically more probable model takes the following facts into account • The output of a neuron depends not only on the current input value, but all the input values over a finite horizon. Thus inputs to neurons are defined over a finite horizon (rather than a single time point). • Synapses are treated as distributed elements rather than lumped elements. Thus synaptic weights are functions defined on a finite support. For the sake of convenience, let the input as well as synaptic weight functions be defined on the support [0, T].

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Novel Mathematical Model of Neuron Let the synaptic weights be wi (t), 1 ≤ i ≤ M i.e. time functions defined on the support [0,T]. Also, let the inputs be given by ai (t),1 ≤ i ≤ M . Thus, the output of the neuron is given by  M  a j (t ) w j (t )  y(t ) = Sign  (9.15)  j =1    More general activation functions ( sigmoid, hyperbolic tangent etc.) could be used. The successive input functions are defined over the interval [0,T]. They are fed as inputs to the continuous time neurons at successive SLOTS. For the sake of notational convenience, we call such a neuron, a continuous time perceptron.

∑

y(t) = Sign[ a1(t)

M

Σ a (t) w (t) ] 1

i

i=1

w1(t) w2(t)

a2(t) .

wm(t)

. . am (t)

Fig 9.1 A novel model of continuous time neuron.

9.5. CONTINUOUS TIME PERCEPTRON LEARNING LAW As in the case of “conventional perceptron”, a continuous time perceptron learning law is given by: (n ) Wi(n+1) (t) = Wi (t) + η ( S(t ) − g(t) ) ai (t)

(9.16)

where S(t) is the target output for the current training example, g(t) is the output generated by the continuous time perceptron and η is a positive constant called the learning rate. The proof of convergence of conventional perceptron learning law, also guarantees the point wise convergence (not necessarily uniform convergence) of synaptic weight functions. Using sigmoid function as the activation function and the continuous perceptron as the model of neuron, it is straightforward to arrive at a continuous time Multi-Layer Perceptron. The conventional back propagation algorithm is generalized to such a feed forward network.

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Modulation Theory: Feed Forward Neural Networks Suppose the synaptic weight functions are chosen as sinusoids i.e. wi (t) = cos υi t or sin υ i t (where υi = 2π f i and fi ’s are frequencies of the sinusoids). The weighted contribution at each neuron actually corresponds to Amplitude Modulation (where the synaptic weight functions are the carrier frequencies and the inputs are the base band signals). We seriously expect that the well known results in Modulation Theory (of communication systems) could be effectively utilized in supervised learning using a single/ multiple layer perceptron.

9.6 SOME IMPORTANT GENERALIZATIONS • Unlike the perceptron model (inputs constitute a vector) discussed previously, it is possible to consider the case where the inputs constitute a three/ multidimensional array (For instance in biological systems, the neurons are indexed by three dimension variables). Utilizing tensor products, the outputs of continuous time neurons are obtained. Also, using the above model of neuron, multi-layer, multidimensional neural networks (such as Multidimensional Multi-layer Perceptron) are designed and studied [Rama1]. • Based on the above model of neuron, it is possible to consider complex valued neural networks in which the input functions, synaptic weight functions, thresholds are complex valued. It is possible to generalize the perceptron learning law, complex valued back propagation algorithm to such complex valued neural networks [Rama1]. • It should be possible to design and study complex valued associative memories based on the above model of neuron.

9.7 SOME OPEN QUESTIONS • Is it possible to generalize Lemma 1 (discussed in section 2) to function spaces? Or equivalently, what is the most general version of Lemma 1 ? • Consider the problem of supervised learning in a function space. Equivalently consider a function space over [0,T]. Define a distance metric over such a space. Design a neural network which can be trained to classify the patterns into finitely many classes (of functions) [Rama 4]. • CLIFFORD NEURAL NETWORKS: Some researchers modeled the neuronal parameters using quaternions. These quaternion based neural networks are utilized in practical applications such as colour night vision [KIM]. Also some authors have utilized geometric algebra in designing novel neural networks. An important open problem is to show that the Clifford/geometric algebra based neural networks have important advantages over the real valued neural networks.

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Multidimensional Neural Networks: Unified Theory

9.8 CONCLUSIONS In this chapter, transforming real valued signals into complex domain (using DFT) and processing them using complex valued neural network is discussed. A novel model of neuron is proposed. Based on such a model, real as well as complex valued neural networks are proposed. Some open research questions are provided.

REFERENCES [AAV] I. N. Aizenberg, N. N. Aizenberg and J. Vandewalle, “Multi-Valued and Universal Binary Neurons”, Kluwer Academic Publishers, 2000. [Hir] A.Hirose, “Complex Valued Neural Networks: Theories and Applications“, World Scientific Publishing Company, November 2003. [KIM] H. Kusamichi, T. Isokawa, N. Matsui et al. “A New Scheme for Colour Night Vision by Quaternion Neural Network“, 2nd International Conference on Autonomous robots & agents, Dec. 13-15, 2004, Palmerston North, New Zealand. [Nit1] T. Nitta and T. Furuya: “A Complex Back-propagation Learning”, Transactions of Information Processing Society of Japan, Vol.32, No.10, pp.1319-1329 (1991) (in Japanese). [Nit2] T. Nitta : “An Extension of the Back-Propagation Algorithm to Complex Numbers,” Neural Networks, Vol.10, No.8, pp.1391-1415 (1997). [Rama1] G. Rama Murthy, “Unified Theory of Control, Communication and Computation”, To be submitted to Proceedings of IEEE. [Rama 2] G. Rama Murthy, “Multi/Infinite Dimensional Neural Networks, Multi/Infinite Dimensional Logic Theory“, International Journal of Neural Systems, Vol.15, No. 3 (2005), 113, June 2005. [Rama 3] G. Rama Murthy and D. Praveen, “Complex-Valued Neural Associative Memory on the Complex Hypercube“, Proceedings of 2004 IEEE International Conference on Cybernetics and Intelligent Systems (CIS-2004), Singapore. [Rama 4] G. Rama Murthy, “Linear Filter Model of Synapses: Associated Novel Real/Complex Valued Neural Networks“, IIIT Technical Report in preparation. [Rama 5] G. Rama Murthy, “Some Novel Real/Complex Valued Neural Network Models,” Proceedings of 9th Fuzzy days, (International Conference on Computational Intelligence), September 2006, Dortmund, Germany, Pages, 473-483.

Advanced Theory of Evolution of Living Systems

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CHAPTER

10

Advanced Theory of Evolution of Living Systems

10.1 UNIFIED THEORY: CYBERNETICS Unified theory of control, communication and computation discovered and formalized by the author led to the birth of the field of Mathematical Cybernetics. Its formal/mathematical basis can be contrasted to the several ad hoc-pseudo mathematical developments in the fields of mathematical biology, psychology, engineering etc. based on the initial enthusiasm generated by Wiener who coined the word. Mathematical cybernetics is the field of formal clarity/completeness which provides abstract models of increasing complexity which demonstrates the equivalence of the functions of control, communication and computation in the machine and the living system from the point of view of the theory/practice (physical hardware) through linear/nonlinear, real/complex dynamical systems. This field of mathematical/formal research on LIVING SYSTEMS in the universe pioneered by the author led him to take a deeper and mathematical approach to the CRUDE, EMPIRICAL THEORY OF EVOLUTION originated by Darwin and seriously investigated by various biological/zoological researchers. It is argued that the concepts utilized by the evolutionists in the theory of organic evolution are incorrect and need to be modified/ updated in view of the unified theory of control, communication and computation. For the purposes of completeness along with brevity some details are provided in the following.

10.2 ORGANIC EVOLUTION Various life forms starting with one/few cell organisms such as amoeba, hydra etc. have evolved from the organic mass in the oceans under certain atmospheric conditions. Some of these organisms have a starting with the ellipsoid based egg. The egg based life forms have formed organs such as eye, mouth due to the organic reactions taking place inside the egg.

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Multidimensional Neural Networks: Unified Theory

With one/two eyes formed on the surface of the egg, due to the rotation of the earth, the natural terrain in the oceans, the egg was constantly drifting in the ocean. The homogeneity of the organic mass has simultaneously led to the formation of several eggs in the same region. This led to the problem of congestion in the region (of eggs). The life forms, in order to cope with the problem began to develop limbs for LOCOMOTION. The remaining organs formed due to the natural environment/atmosphere have similar topological features in different species of living systems (UNIFICATION of various LIVING species). The life forms began to jostle and to handle the environmental needs, the ellipsoid based egg deformed to form various shapes for the body and primitive organs (non-intelligence based). These differences in the shape/topological features of the body, organs led to the classification of such living systems into species such as frogs, fish, crocodiles etc. The initial organic mass based life had no intelligence. The set of characteristics that are common to various life forms have formed over a large length of time. Some novel and innovative concepts which are the distinguishing features of this advanced theory are briefly described below.

10.3 EVOLUTION OF LIVING SYSTEMS: INNOVATIVE PRINCIPLES (A) Principle of Equivalence of Trainability of Intelligence in all Natural Living Systems: The observed variation of intelligence in living systems could be due to variation of the biochemical content in the brain. But by training various living systems to learn a language, they could be made intelligent and a certain living animal culture with intelligence can be developed. In essence, various lower/higher level animals could be organized in a zoo and rendered useful to themselves as well as homo-sapiens. (B) Principle of Non-Necessity of Perceived Needs of Living Systems: Various species characteristics as needs have evolved over a period of time. These cravings/needs are genetically replicated. Some of these needs/cravings are not necessary to sustain life. For instance, METABOLISM which leads to killing of one life form by another was an accident and is not necessary to sustain the life of various species of organic life based machines. (C) “Life” and “death” are identical to functioning and non-functioning machines. It should be possible to take an organically non-decayed species form which is “dead” due to “bleeding”, heart failure, some organic decay, malignant growth etc. and make it living. Thus, in some sense, there is no death. In summary, unified theory sheds a proper light on the previously less understood concepts of death and life. (D) Reproduction, as a species need has evolved over a long period of time. This has led to the problem of overpopulation in some parts of the planet. As is well known, reproductive appetite could be turned off.

Advanced Theory of Evolution of Living Systems

139

(E) Various problems of homo-sapiens in individual families, in communities, in geographical boundaries, across countries have arisen due to the programming of the homo-sapien machines. For instance, the tradition of “battle” in European communities is a disease. Thus, various human, living machine diseases are cured through proper understanding like problems in science or mathematics. (F) By cross fertilizing the eggs of different species of living machine such as monkeys, homo-sapiens, lions, tigers etc. it might be possible to give birth to lower-higher animal, sea-land animal, air-land animal, other combinations of living machines. Thus, creation/organization/civilization/culturing of whole new class of living machines is a promising possibility. Utilizing the mathematical theories of optimization (of various types...functional, multivariate, constrained etc), MORPHOGENESIS etc. various macro-scale aspects of living systems—shapes of body organs, their functions etc; micro-scale aspects of living systems—structure of DNA, protein chains, structure of genes etc. are formally explained. Thus, systems in living as well as non-living material universe are endowed with an optimization interpretation (at micro-scale as well as macro-scale).

10.4 CONCLUSIONS In an effort to understand non-living physical reality, various sub-fields of science such as physics and chemistry were developed. Based on experimental observations from physical reality, various mathematical, empirical theories were constucted to derive laws of nature. These laws, principles, theories on non-living physical reality were utilized to develop science and engineering. The field of biology was developed to understand the composition, operation, coordination of various organs/functional units of living systems in nature such as homo-sapiens, tigers etc. The distinction between living pysical reality and non-living physical reality was very puzzling to scientists. In the mid 1940’s, N. Wiener coined the word CYBERNETICS for the field dedicated to understand the control, communication and computation functions of living systems. The author pioneered the field of mathematical cybernetics by unifying the control, communication and computation functions of living system functional units. Thus, a mathematical model of natural living systems was developed. It is shown that in the context of one dimensional linear dynamical systems that the unification includes various other functions alongwith control, communication and computation. By utilizing the tensor state space representation of certain multi/infinite dimensional linear dynamical systems discovered by the author, cybernetics results for multi/infinite dimensional systems were developed. These results enabled the development of multi/infinte dimensional coding, computation and system theories. The author also made some pioneering investigations into the functions of various natural living sytems. These investigations provided the important conclusion that the living machines such as homo-sapiens, tigers etc. programmed themselves for functions

140

Multidimensional Neural Networks: Unified Theory

such as metabolism, sex etc. Many issues of importance to the living machines such as control/coordination of them, diseases, programmed bad habits are all addressed based on a proper understanding of the theory. The advanced theory of evolution resulting from the unified theory of control, communication and computation resulted in new perspectives into nature based living systems. In summary, in this book, the author related multidimensional logic, coding and control theories to the concept of multidimensional neural networks (proposed by him). He innovated a novel complex signum function and proposed a novel complex valued associative memory. Several novel models of neuron are proposed and associated real as well as complex valued neural networks are discussed.

Index

141

Index A A codeword 91 A Human/Animal Brain 125 A Multi-Layer Feed Forward Network 120 A Sigmoid Function 120 Astable State 16, 124 Abstract Mathematical Structure 118 Abstract Model 79 Abstract models 137 Accurate 118 Activation Function 118, 125, 133, 134 Activation Functions 125 Adaptive Neural Networks 24 Addition 120 Additive 42 Additive I.I.D. Noise Term 72 Adjoint Equations 87 Admissible Control Tensors 87 Admissible Functions 64 Admissible Sequence 83, 108, 109, 113 Advanced Theory of Evolution 137 Algebraic Geometry 45 Algebraic Threshold Function 1, 29, 95, 97 All One Dimensional Logic Gates 81 All-Ones Tensor 40 Amplitude Modulation 135 Analog Linear Filter. 121 Analysis 62, 63, 76 AND, OR , NOR, NAND, XOR Gate 17 AND, OR, NOR, NAND, XOR Gates 81 AND, OR, NOR, NAND, XOR, NOT Gates 90 AND, OR, NOT, NAND, XOR, NOR 9 Animal 1 Approximation 55 Approximation Theory 55 Arbitrary Algebraic Threshold Function 12

Arbitrary Open 54 ARMA Time Series Model 72 Array 35 Artificial 3, 107 Artificial Neural Networks 81, 117, 118, 129 Associated Boundary Conditions 87 Associative Memory 79, 80, 125, 129 Attasi’s Model 64 Audio Signal 129 Auto-Regressive (AR) 70 Auto-Regressive Moving Average (ARMA) 70 Autocorrelation Tensors 71 Automata 4, 10

B Back Propagation Algorithm 118, 122, 126, 130, 133 Basic Model of a Neuron 129 Basis 33 Behavior 64, 96 Behavior Approach 64 Better Model of Neurons 117 Bijection 131 Bijective Linear Transformation 131 Binary Arrays 9 Binary Codes 50 “Binary Filtering” 113 Binary Linear Multidimensional Code, 42 Binary Tensor 33 Binary Valued Functions 123 Binary Vector 123 Biological Neural Networks 117 Biological Neurons 117 Biological Systems 135 Bipartite Graph 104 Block Codes 107 Block Symmetric Tensor 18

Multidimensional Neural Networks: Unified Theory

142

Block Tensor 32 Block Tensors 33 Blocked Tensor 32 Boolean Algebra 3 Boolean Function 46, 48, 49 Boolean Functions 9, 46, 47 Boolean Logic Theory 80 Bounded Extent 113 Bounded Lattice 24, 53, 54 Bounded Lattices 54 Bounded Magnitude 113 Bounded Support [0, T] 118, 123 Bounded Support Input 113 Brain of Powerful Robots 80

C Causal/Non-Causal Parts 75 Certain Discrete Time Multidimensional Systems 83 Certain Discrete Time System 82 Certain Multidimensional Linear System 109 Certain Multidimensional Linear Systems 62 Certain Multidimensional System 82 Certain Multidimensional Systems 69, 82 Certain Multi/Infinite 65 Channels 107 Characteristic Tensor 32 Cholesky Decomposition 56, 57 Circuits 32 Civilization, 1 Class of Input Signals 118 Class of Inputs 123 Class of Problems in P? 56 Classes (of Functions) 135 Classes of Functions 55 Classes of NP-Hard Problems 56 Clifford Neural Networks 135 Clifford/Geometric Algebra 135 Code 29, 49 Codes 27 Codeword 33, 38, 40, 81 Codeword Array 35 Codeword Tensor 35, 42, 43, 46, 47, 49 Codeword Vector 35 Codewords 27, 53, 54

Coding 4 Coding Theory Approach 107 Coding Theory 27 Colored Noise 71 Colored Noise Model 72 Common Thread 80 Common Thread of Neural Networks 80, 81 Communication 4, 79, 80, 81 Communication and Computation 91 Communication Systems 135 Commutative Operator 113 Compact 54 Compact Set 53, 54, 55 Compatible Tensors 15, 59 Complex 97 Complex Domain 136 Complex Hypercube 96, 97 Complex Neural Networks 24 Complex Number 97, 120 Complex Part of the Weight Matrix 103 Complex Signum Function 105, 125 Complex Synaptic Weight Matrix 102 Complex Synaptic Weights 130 Complex Valued 96 Complex Valued Associative memory 96, 102, 125 Complex Valued Backpropagation Algorithm 135 Complex Valued, Continuous Time Associative Memory 124 Complex Valued, Continuous Time Multi-layer Perceptron 126 Complex Valued, Continuous Time Neuron 125 Complex Valued Inputs 129 Complex Valued Multi-Layer Perceptron 133 Complex Valued Neural Net 102 Complex Valued Neural Network 96, 101, 105, 133, 135, 136 Complex Valued Neural Networks 118, 125, 130 Complex Valued Neuron 125 Complex Valued Perceptron 133 Complex Valued Perceptrons 133 Complex Valued Sigmoidal Function 126 Complex Valued Synaptic Weights 129 Complex Valued Vector 97 Complex-Valued 97, 135 Complexity Theory 55 Computation 3

Index

Computation 4, 79, 80, 81, 90 Computation Functions 92 Concept of a Logic Gate 80 Concept of Error 81 Concept of Error Correcting Code 80 Concept of Neural Network 81 Concepts 82 Connection Matrix 57, 58, 96, 113 Connection Structure 12, 14, 18, 21, 24, 27, 29, 31, 32, 34, 92 Connection Tensor 22 Connectionist Structure 11, 13 Constrained Static Optimization 53 Constraint 57 Constraint set 24, 53, 54, 55, 57, 84 Continuous Time 86 Continuous Function 55 Continuous Index/Argument 87 Continuous Objective Functions 55 Continuous Time 62, 71 Continuous Time Associative Memory 122 Continuous Time I. I. D. Noise 72 Continuous Time Linear Multidimensional Systems 62 Continuous Time Linear System 123 Continuous Time Multi-Layer 134 Continuous Time Neural Associative Memory 91 Continuous Time Neural Networks 24, 121, 126 Continuous Time Neuron 121 Continuous Time Neurons 119, 135 Continuous Time or Discrete Time 4 Continuous Time Perceptron 118, 134 Continuous Time Perceptron Learning Law 119, 134 Continuous Time Perceptron Law 118, 134 Continuous Time Perceptrons 121 Continuous Time Strucured Markov Random field 71 Continuous Time System 124 Continuous Time Systems 66 Continuous Time Version 75 Continuous Time Versions 86 Continuous Time Versions of These Models 71 Continuous-time 118, 129 Continuous-time Multi-Layer Perceptron 120, 121 Continuous-Time Neuron 120 Contraction 12, 14, 35, 48

143

Control 3, 4, 79, 80, 81 Control, Communication 4, 90, 92 Control, Communication and Computation 92 Controllability 61, 62, 82 Controllability, Observability, Stability 69 Conventional Approaches 68 Conventional Back Propagation Algorithm 134 Conventional Model of Neuron 133 Conventional Perceptron 134 Conventional Perceptrons 133 Conventional State Space Models 77 Converge 97 Converge Uniformly 119 Convergence 22 Convergence of Energy 16 Convergence Properties 96 Convergence Theorem 10, 14, 17, 21, 30, 125 Convergence Theorems 96 Converges 133 Convex Polygon/Polytope 57 Convex Polytope 57 Convolution 113, 118, 120 Convolution operation 119 Correcting codes 81 Corrupted codeword 27 Corrupted word 92 Cosets 41 Cost function 84, 109 Criterion 108 Cut 30, 32 Cut codes 33 Cut space 32, 33, 92 Cuts 32 Cybernetics 79 Cycle 21, 97 Cycle of length 2, 97 Cycle of length two 104 Cycles 22, 30 Cycles in the State Space 20

D Darwin 137 Data bases 62 “Death” 138 Decoders 34

144

Decoding 27, 45 Decoding Algorithm 45 Decoding Techniques 28, 52 Decomposition Principle 29, 54, 55, 57 Decompositions of the State Space 76 Definition 23 Degree n 11 Derivative 67 Design 63 Design a Constellation 27 Design, Analysis 61, 62 Design of Codes 28 Determinant of the Matrix 132 DFT 133, 136 Difference Equation 71 Difference Equations 61 Difference in Energy 16 Differential Equations 61, 69 Differential/partial differential equations 76 Digital filter 122 Digital filters 130 Digital Signal Processing 130 Dimension 13, 22, 29 Dimensional 27 Dimensional Continuous Time 67 Dimensional Dynamical Systems 65 Dimensional Hypercube 53 Dimensional Linear as well as Non-Linear Codes 34 Dimensional Neural Networks 27 Discovered and Formalized 82 Discovery and Formalization 80 Discrete Fourier Transform 130 Discrete Fourier Transform matrix 133 Discrete Fourier Transformation 132 Discrete Maximal Principle 108 Discrete maximum principle 83 Discrete memoryless channel 37 Discrete Sequence of Samples 132 Discrete Sequences 130 Discrete Time 82 Discrete Time Input Signals 122 Discrete Time Linear System 124 Discrete Time Multidimensional Neural Network 29 Discrete Time Multidimensional Systems 62 Discrete Time Multi/Infinitedimensional System 83

Multidimensional Neural Networks: Unified Theory

Discrete Time Simulation 124 Discrete Time System 123, 124 Discrete Time Systems 66, 67, 82 Discrete Time, Time Varying Linear Systems: 83 Discrete Time, Two Dimensional System 62 Discretizing a Continuous Time System 124 Disease 139 Distance Measures 42 Distance Metric Over Such a Space 135 Distributed Dynamical Systems 73, 76, 77 Distributed Elements 117, 126, 133 Distributed Nature of 74 Dual 29 DUAL of Signal Design Problem 113 Dynamic as well as Static Linear Systems 76 Dynamic Optimization 24 Dynamic Programming Principle 83 Dynamical System 29, 66 Dynamical Systems 63, 75, 86, 137 Dynamics 82

E Each Neuron Act 135 Egg 138 Eigentensors 72 Eigenvalue 58 Eigenvalues 72 Eigenvectors 58 Electrical Transmission Lines 73 Ellipsoid 138 Ellipsoid Based Egg 137 Empirical Theory of Evolution 137 Encode 35 Encoded Codeword 37 Encoder 4 Encoders 34 Encoding 27, 35 Encoding Algorithm 45 Encoding Procedure 37, 38, 43, 44 Encoding/Decoding Algorithms 35, 45 Energy Density Matrix 123 Energy density tensor 86, 89, 124 Energy E 16 Energy Function 10, 1, 15, 16, 17, 21, 27, 28, 29, 30, 31, 38, 43, 44, 45, 58, 59, 81, 90, 91, 95, 97, 98, 102, 103, 105, 124

Index

145

Energy Function (Quadratic) 34 Energy Function Being A Quadratic Form 91 Energy Function Hill Climbing 124 Energy Functions 18, 19, 20, 24, 38, 45, 51, 54, 90 Energy Landscape 27 Energy Values 21 Entropy 36 Entropy/Uncertainty 37 Equilibrium Distribution 72 Error Correcting Codes 31, 34, 59, 81 Errors 27, 39 Euclidean Space 117 Every Codeword 91 Every Local Maximum 81, 91 Evolution 62 Evolution At Node 14 Evolution Equations 61 Evolution of The System 74 Evolutionists 137 Exclusive OR 35, 129

F Fast Fourier Transform 130 Feed Forward Network 118, 134 Feed-forward/Recurrent Networks of Neurons 118 ‘Field’, F 121 Filter 118 Filtering 73, 107 Finite Dimensional Vector Space 63 Finite Field 35, 42 Finite Fields 59 Finite Impulse Response Filter 122 Finite Impulse Response Model 121 Finite Impulse Response Model of Synapse 118 Finite Support 118, 120, 123 Finitely Many Classes 117 Fourier Laplace Transform 121 Fully Parallel Mode 14, 16, 20, 96, 103, 104 Fully Symmetric Connection Tensor 91 Fully Symmetric Tensor 10, 14, 29, 30, 82, 84 Fully Symmetric Tensor of 22 Fully Symmetric Tensor S 13, 15 Function 31 Function Space 118, 120, 121

Function Spaces 135 Fundamental 4

G G/M/1-Type Structure 72 Game-Theoretic Codes: Optimal Codes 39 Generalization of Back Propagation Algorithm 120 Generalized Logic Circuit 19 Generalized Logic Function 19 Generalized Logic Gate 19 Generalized Multidimensional Logic Gate 19 Generalized Multidimensional Neural Network 19 Generalized Neural 91 Generalized Neural Network 11, 28 Generalized Neural Networks 124 Generalized/Multidimensional Neural Networks 12 Generator Tensor 29, 34, 35, 37, 38, 40, 42, 49, 53, 71, 72, 90 Generator Tensor, Codeword Tensors 52 Generator Tensor G 43 Generator Tensors 35 Generator/Information Tensor 36 Generator/Parity Check Tensor 92 Generator/Parity Check Tensors 36 Global Maximum 28, 29, 30, 34, 38, 40, 81 Global Optimization 55 Global Optimum 28, 92 Global Optimum Control Vector 108 Global Optimum Impulse Response 115 Global Optimum Stable State 52 Global States 63 Global/Local Optimum 54 Graph-Theoretic Code 22, 57, 92 Graphoid 29, 31, 33, 34, 92 Graphoid Based Codes 29 Graphoid Codes 31, 34 Graphoid Theoretic Codes 32 Graphs of Convergence 102 Group 42, 52

H Hadamard Matrix 46, 47 Half Plane Causal 75

Multidimensional Neural Networks: Unified Theory

146

Half Plane Causality 70 Half Plane, Quarter Plane Causal Type Neighbourhood 76 Hamiltonian 83, 87 Hamming Distance 33, 38, 42, 43, 58 Hardware 5 Heine-Borel Theorem 53 Hermitian 97 Hermitian Symmetric Matrix 24 Hermitian Synaptic Weight Matrix 97 Heuristic Procedures 59 Higher Degree 23 Higher Degree Forms 19 Higher Dimensional Space 75 Hirose 105 Homo-sapien 1, 2 Homo-sapien Machines 139 Homo-sapiens 2, 129, 138 Homogeneous 71, 74 Homogeneous Form of Degree N. 11, 12 Homogeneous Quadratic Form 11 Homogeneous Stochastic Linear Systems 71 Honig 123 Hop-skip Algorithm 57 Hopfield 80, 102, 103, 129 Hopfield Associative Memory 96, 103, 129 Hopfield Convergence Theorem 100 Hopfield Model 95, 96 Hopfield Network 124 Hopfield Neural Network 81, 115 Hopfield/Amari 79 Hypercube 23, 58, 84, 95 Hypercubes 57 ‘Hyperplane’ 121 Hypersphere 58 Hyperspheres 54

I Identity Tensor 40 IDFT 133 Image Models 73 Image Processing 63, 96, 105 Image Processing, Tomography 76 Images 129 Imaginary Parts of 97

Impulse Response 61, 113, 117, 121 Impulse Response of FIR Filters 122 Impulse Response Sequences 122 Impulse Response Tensor 83, 85, 86, 89 Impulse Response Values 113 Impulse Response 113 Incidence Tensor 32 Independent, Identically Distributed Noise Model 72 Infinitedimensional 20, 23 Infinitedimensional Code 36 Infinitedimensional Codes 29, 36 Infinitedimensional Hypercube 23 Infinitedimensional Linear System 109 Infinitedimensional Logic Function 22, 23 Infinitedimensional Logic Theory 10, 20, 22 Infinitedimensional Neural Network 22 Infinitedimensional Symmetric Matrix 20 Infinitedimensional Systems 66 Infinitedimensional Tensor 36 Infinitedimensional Vector 21 Infinitedimensional Vector Space 20 Infinite Impulse Response Filter 122 Infinite Order/Dimension 29 Infinitedimensional Logic Circuit 23 Infinitedimensional Logic Gate 22 Infinitedimensional Logic Gates 23 Infinitedimensional Logic Synthesis 23 Information Tensor 35, 37, 38, 46, 49, 52 Information Theory 107 Information Vector 35 Initial Condition 83 Initial State Tensor 72 Inner and Outer Products Make Sense 84 Inner Product 14, 15, 38, 49, 53, 59, 69, 71, 84, 109 Inner Product of the Given Tensors 12 Inner Product Operation 82 Inner Product Operator 15 Inner Product/Outer Product 70 Inner/Outer Product 72 Innovative Ideas 29 Input 113, 117 Input and Output Signal States 17, 81, 90 Input Coupling Tensor 67, 68, 82 Input Energy Density Matrix 113 Input Functions 135 Input, Output, State Variables 66

Index

147

Input Patterns 117, 122 Input Sequence 109, 122 Input Signal Samples 117 Input Signal Tensors 83 Input Signals 108, 118 Input Tensor 42, 67, 75, 82, 87 Input-Output Models 63 Input-Output Point of View 63 Input-Output Representations 61 Input/Stable States 10 Inputs 117 Inputs to Neurons 133 Integer Programming 52 Integer/Non-linear Programming 59 Integral of Tensor Function 86 Intelligence 3, 138 Interconnection Structure 23 Interesting Observation 97 Internal Representations 64 Invariant Distribution 72 Invariants of a Tensor 17 Invertible 133 Isolated Theories 79 Isolating 97

J Jacobian Matrix 76 Jacobian Tensor 76

K Knapsack Problem 57

L Language 138 Latent Variable Models 64 Lattice 22, 50, 59 Lattice (Unbounded Lattice) 24 Lattice 5 Learning Laws 117 Learning Rate 119, 134 Lee Distance 42, 43, 44, 45 Lee Weight 44 “Life” 138

Light Wave 34 Lindeloff’s Covering Lemma 54 Linear 66, 118 Linear Algebra 61, 63 Linear Algebra Concepts 35 Linear As Well As Non-linear Codes 90 Linear Block Code 29, 81 Linear Block Codes 45, 57 Linear Block Multidimensional Code 28 Linear Block Multidimensional Codes 53 Linear Code 113 Linear Discrete Time Filter 108 Linear Dynamical System 88 Linear Dynamical Systems 70, 91 Linear Equations 47 Linear Filtering 113 Linear Filters 126 Linear Multidimensional 49 Linear Multidimensional Block Code 38, 41, 42 Linear Multidimensional Block Codes 38, 41 Linear Multidimensional Codeword Tensor 49 Linear Operator 63, 70, 72 Linear Operators 62, 109 Linear Prediction 73 Linear Programming 56 Linear Programming Problems 57 Linear Programming Problems: Decomposition Principle 57 Linear Separability 121 Linear Space 68 Linear System 107, 117 Linear System Theory 61 Linear Systems 5, 24, 61 Linear Systems/Filters 118 Linear Tensor/Vector 32 Linear Tensor/Vector Space 32 Linear Time Invariant Continous Time System 81 Linear Time Invariant Multidimensional System 89 Linear Time Varying Multidimensional System 91 Linear Time Varying Multi/Infinitedimensional Dynamical Systems 84 Linear Time Varying System 108 Linear Time Varying Systems 112 Linear Time-invariant System 124 Linear Transformation, 131 Linear Transformation Groups 51

Multidimensional Neural Networks: Unified Theory

148

Linear Transformations 65 Linear/Non-linear 137 Linear/Non-linear Codewords 54 Linear/Non-linear Dynamical Systems 62 Linear/Non-linear Multidimensional Codewords 54 Linearly Separable 122, 131 Linearly Separable Patterns 131 Linearly Separable Test Patterns 133 Linearly Seperable 132 Living Systems 4, 137 Living Systems/Machines 129 Local Control 62, 66, 76 Local Input 66 Local Input, Local Output 76 Local Maxima 100 Local Maximum 28, 29, 91 Local Minimum/Maximum Of 17 Local Optima 55, 57, 81, 90 Local Optima of Energy Function 19, 81 Local Optima of The Energy Functions 18 Local Optimum 19, 54, 91, 92 Local Optimum Control Vector 108 Local Output 66 Local State 62, 66, 76 Local State, Local Control 76 Local State Tensor 75 Local State, The Input 64 Local States 63 Locomotion 138 Logic Circuit 10, 19 Logic Circuits 10, 19, 80 Logic Function 10 Logic Functions 17, 22, 114 Logic Gate 10, 81 Logic Gates 3, 9, 19, 22, 80 Logic Gates, Logic Circuits 17 Logic Synthesis 9, 10, 80 Logic Theory 9, 81 Lumped 117 Lumped Elements 117, 126, 133 Lyapunov Function 124 M m-d Hopfield Neural Network 82 m-d Neural Network 114 M-Dimensional Euclidean Space 131

M-Dimensional Input Array 19 M/G/1-type Structure 72 Machines 3, 4 Macro-scale Aspects 139 Magnetic and Optical Recording Systems 81 Magnetic/Optical Recording Channel 123 Magnetic/Optical Recording Channels 122 Markov Chains 71 Markov Random Field is a Stochastic Linear System 73 Markovian 64 Markovian Property 65 Markovian Source 37 Mathematical Abstraction 10 Mathematical Clarity 80 Mathematical Cybernetics 137 Mathematical Model 11, 12, 13, 79, 90 Mathematical Structure 120 Mathematical Theory of Communication 36 Matrix 133 Maximization of a Quadratic Form 52 Maximization of Multivariate Polynomial 37 Maximum Eigenvector 58 Maximum Likelihood Decoding 28, 29, 33, 34, 41, 43, 44, 45, 50, 57, 59, 81, 92 Maximum Likelihood Decoding Problem 37, 42, 43 Maximum Likelihood Problem (MID) 41 Maximum of The Quadratic Energy Function 34 Maximum Weight Independent Set 52 Maximum/Minimum 55 Maximum/Minimum Energy States 90 McCulloch and Pitts 118 McCulloch-Pitts Neuron 95 Mean Square Error 107 Measurement Noise 73 Measurement Noise Models 72 Message 4 Metabolism 138 Metabolism and Reproduction 1, 6 Metric Space 54 Minimum 33 Minimum Cut 28, 29, 31, 34, 56 Minimum Cut Computation 56 Minimum Cut Problem 52 Minimum Distance 34, 38, 39, 45 Minimum Distance, Correctable Errors 28 Minimum Weight 31

Index

Minimum/Maximum Energy States 10, 17 Minsky 118 Mode of Operation 15 Model 96, 97 Model of Associative Memory 129 Model of Neuron 118 Model of Synapse 122 Modeling, Design 62 Modeling Distributed Dynamical Systems 74 Models 117 Models of Distributed Systems 74 Models of Neuronal Circuitry 129 Models of Neurons 122 Models of Tomographic Images of Brain 73 Modern Approaches 68 Modes of Operation 14, 96 Modes, Serial 96 Modulation Theory: Feed Forward Neural Networks 135 Monomial 37 Monomials 47, 53 More General Constraint Sets 53 Morphogenesis 139 Multi Layer Perceptron 122 Multidimensional 66, 90, 110 Multidimensional Array/Tensor 23 Multidimensional Arrays 10, 36 Multidimensional Block Code 90 Multidimensional Bounded Lattice 19, 54 Multidimensional Channel 33, 36, 43 Multidimensional Code 33, 35 Multidimensional Codes 27, 28, 34, 51, 59 Multidimensional Codeword 44, 45 Multidimensional Codeword Set 54 Multidimensional Codeword Tensors 91 Multidimensional Codewords 38 Multidimensional Coding Theory 27, 90 Multidimensional Communication Systems 36 Multidimensional Constrained Optimization Problem 55 Multidimensional Decoding Techniques 53 Multidimensional Discrete Time Dynamical System 67 Multidimensional Encoder 19 Multidimensional Encoders As Well As Decoders 28 Multidimensional Encoding Scheme 37

149

Multidimensional Error Correcting Code 27, 31 Multidimensional Error Correcting Codes 28 Multidimensional Generalization 56 Multidimensional Generalizations 126 Multidimensional Generalized Neural Networks 90 Multidimensional Graph-type Structure 92 Multidimensional Hopfield Neural Network 80 Multidimensional Hypercube 17, 19, 28, 47, 91 Multidimensional Information Array (Information Theory) 35 Multidimensional Information Theory 36 Multidimensional Lattice 19, 23, 27, 28, 38, 51, 53 Multidimensional Linear As Well As Non-linear 80 Multidimensional Linear Block Code 28, 33, 37, 38, 39, 40, 43 Multidimensional Linear Block Codes 29, 34 Multidimensional Linear Code 37, 90, 92 Multidimensional Linear Codes 37, 54 Multidimensional Linear Codeword Constellation 34 Multidimensional Linear Space 69, 70 Multidimensional Linear Spaces 62 Multidimensional Linear Systems 5 Multidimensional Logic 19 Multidimensional Logic Circuit 18, 23 Multidimensional Logic Function 17, 18 Multidimensional Logic Functions 10, 18, 19, 25, 90 Multidimensional Logic Gate 10, 17, 19, 90 Multidimensional Logic Gate Functions 17, 90 Multidimensional Logic Gate/Circuit 10 Multidimensional Logic Gates 18, 23 Multidimensional Logic Synthesis 18, 25 Multidimensional Logic Theory 17, 22, 90 Multidimensional Logic Theory/Logic Synthesis 23 Multidimensional Logic Units 10 Multidimensional Metric Space 53 Multidimensional Neural Network 10, 11, 13, 14, 15, 18, 22, 27, 29, 30, 31, 34, 59, 90, 92, 118 Multidimensional Neural Networks 10, 11, 18, 23, 28, 51, 59, 80, 82, 90, 92 Multidimensional Neuron 15 Multidimensional Neuronal Element 29 Multidimensional Neurons 13 Multidimensional Non-Binary/Binary Codes 53 Multidimensional Optimization Theory 53 Multidimensional Space 65 Multidimensional Stochastic Dynamical System 72

150

Multidimensional System 58, 65, 73 Multidimensional System Theory 62 Multidimensional Systems 62, 63, 66, 71, 80, 82, 90 Multidimensional Tensor Codeword 40 Multidimensions 114 Multi-input,Multi-Output (MIMO) 124 Multi-input, Multi-output (MIMO) Channels 108 Multi-input, Multi-output (MIMO) Linear Time Varying Filters 123 Multi-Layer Feed Forward Neural Network 129 Multi-layer, Multidimensional Neural Networks 135 Multi-Layer Neural Networks 117 Multi-Layer Perceptron 117, 118, 129 “Multi-Order” System 66 Multi-Tensor Variate Polynomials 52 Multi-Variate Polynomial Equations 52 Multi-Variate Quadratic Form 92 Multi/Infinitedimensional 5, 63 Multi/Infinitedimensional Coding Theory 35, 62 Multi/Infinitedimensional Distributed Systems 63 Multi/Infinitedimensional Hypercube 86, 87 Multi/Infinitedimensional Linear Codes 59 Multi/Infinitedimensional Linear System 87 Multi/Infinitedimensional Linear Systems 63, 70 Multi/Infinitedimensional Neural Networks 10, 62 Multi/Infinitedimensional State 65 Multi/Infinitedimensional State Space 75 Multi/Infinitedimensional State Space Structure 70 Multi/Infinitedimensional Structured Markov Random Field 71 Multi/Infinitedimensional System 63 Multi/Infinitedimensional System Theory 62, 68, 69 Multi/Infinitedimensional Systems 63, 69, 84 Multi/Infinitedimensional Versions of Time-series Models 70 Multi/Infnintedimensional State Space 70 Multiple Arguments 74 Multiplication 120 Multiplication of Tensors 12 Multiplicative Group 42 Multiplicative Representation 42 Multivariate Polynomial 29, 52, 53, 54, 59 Multivariate Polynomials 28, 29, 34, 51, 52, 54

Multidimensional Neural Networks: Unified Theory

N Natural Linear 107 Natural Living Systems 138 Nearest/Farthest Neighbourhood Set 75 Necessary Condition 108, 110 Neighbourhood Sets 75 Nerode Equivalence 63, 64 Networks 91 Neumann Series 124 Neural 91 Neural Net 102, 104 Neural Network 28, 30, 56, 57, 58, 90, 95, 97, 108, 110, 113, 122, 135 Neural Network Model 27 Neural Networks 59, 120, 121, 130, 133, 135 Neural/Generalized Neural Networks 38, 59 Neuron 12, 13, 122 Neuron Output 96 Neuronal Element 29 Neuronal Models 117, 118, 120 No Natural Notion of Causality 82 Node 13, 99 Nodes 13 Noise Model 72, 73 Noise Process 73 Noise Processes 72 Noise Terms 72 Noisy Communication Channels 107 Non-binary Codes 28, 29, 50 Non-binary Linear Codes 41 Non-causal Two Dimensional Dynamics 65 Non-homogeneity 74 Non-linear 28 Non-linear Block Codes 45 Non-linear Codes 27, 46, 47, 90 Non-linear Multidimensional Codes 29, 46, 50 Non-linear System 66 Non-linear Systems 107 Non-linearly Separable Patterns 118, 129, 133 Non-negative Diagonal 104 Non-planar Graph 31 Non-stationary Fields 74 Non-stationary Tensor Fields 75 Nonstationary Fields 74 Novel Associative Memory 122

Index

151

Novel Continuous Time Associative Memory 118 Novel Model of a Neuron 133 Novel Model of Associative Memory 126 Novel Model of Continuous Time Neuron 130 Novel Model of Neuron 130, 136 Novel Models of Neurons 126 NP-hard Problem 52, 56, 57 NP-hard Problems 55, 59

O Objective Function 55, 57, 80, 87, 91 Objective Function J 109 Objective Functions 55 Observability 61, 62, 82 One Dimensional 5 One Dimensional Arrays 9 One Dimensional Arrays i.e.vectors 90 One Dimensional Arrays of Zeroes and Ones 80 One Dimensional Coding Theory 35 One Dimensional Error Control Coding Theory 34 One Dimensional Error Correcting Code 81 One Dimensional Error Correcting Codes 90 One Dimensional Linear Dynamic Systems 63 One Dimensional Linear Space 69 One Dimensional Linear System 82 One Dimensional Linear Systems 69 One Dimensional Logic Functions 18, 81 One Dimensional Logic Theory 17, 18, 50, 51, 80, 90 One Dimensional Neural Network 11, 20, 81 One Dimensional Neural Networks 20, 80, 81 One Dimensional Non-Linear Codes 46 One Dimensional Optimal Control Vectors 81 One Dimensional Stochastic Linear Systems 71 One Dimensional System Theory 76 One Dimensional Systems 71, 80, 82 One-dimensional Linear Dynamical System 108 One/Two/Three Dimensional Information 129 Open Problem 83 Open Questions 130 Open Research Problem 107, 122, 123 Open Set 54 Open/Closed Sets 29 Operating in the Fully Parallel Mode 97 Optical Networks 34

Optimal Binary Filters: Neural Networks 107 Optimal Code 92 Optimal Codeword 92 Optimal Codeword Vector 81 Optimal Control 5, 92, 110 Optimal Control of Certain Multidimensional System 82 Optimal Control Problem 87, 114 Optimal Control Sequence 84 Optimal Control Tensor 82, 84, 86, 89, 91, 92 Optimal Control Tensor Sequence 83 Optimal Control Tensors 80, 82, 87, 91, 92 Optimal Control Vector 81, 112, 123, 124 Optimal Control Vectors 81, 92, 108 Optimal Control/ Signal Design 115 Optimal Filter 107 Optimal Filter Design Problem 113 Optimal Filter Problem 113 Optimal Filtering Problem 107, 114 Optimal Input 113 Optimal Input Vector 123 Optimal Linear Multidimensional Code 91 Optimal Logic Functions 92 Optimal Logic Gate Output 81 Optimal Multidimensional Logic Functions 91 Optimal Sequence 109 Optimal Set of Impulse Responses 113 Optimal Signal Design 107 Optimal Switching Function 91, 92 Optimality Condition 83 Optimization 23, 27, 28, 54, 55, 59, 80 Optimization Approach 80 Optimization Constraint 27 Optimization of Multivariate Polynomials 28, 50 Optimization of Quadratic/Higher Degree Forms 28 Optimization Over More General Constraint Sets, 54 Optimum Input Signal 108 Optimum Stable State 56 Order 13, 22, 29 Ordinary Difference 69 Ordinary/Partial Difference/Differential Equations 76 Organic Evolution 137 Organic Life Based Machines 138 Organic Mass 137, 138 Organically Non-decayed 138 Oscillate 104

Multidimensional Neural Networks: Unified Theory

152

Oscillates 105 Oscillation 105 Outer Product 14, 35, 69 Outer Product of Tensors 12 Output 64, 113 Output Generated 119 Output of A Neuron 133 Output of Each Synapse 118 Output States 10 Output Tensor 67, 68, 82 Output Tensors 72 Outputs 117

P Parallel Computers 9 Parallel Data Transfer 34 Parallel Mode 21, 22, 97 Parallel Modes 14 Parity Check Equations 35 Parity Check Matrices 35 Parity Check Matrix 35 Parity Check Tensor 29, 40 Partial Differential Equations 68, 69 Pattern Recognition 96 Patterns 132 Perceptron 118, 119, 129, 134 Perceptron Learning Law 118, 135 Perceptron Model 135 Planar Graphs 32 Plant and Measurement Noise 73 Plant Noise 72 Plant Noise Model 73 Point Wise Convergence 134 Polynomial 46, 49 Polynomial Representation 37, 40 Polynomial Time Algorithms 57 Polynomials 46, 47 Polynomials, Power Series 55 Pontryagin Function 83, 87 Positive Definite Symmetric Matrix 56, 57 Positive Definite Synaptic Weight Matrix 56 Power Spectrum 71 Preciseness 80 Prediction 73 Prime 43

Primitive Organs 138 Problem of Communication 79 Problem of Computation 79 Problem of Control 79 Procedures 27 Product 15 Programming 139 Programming Problem 59 Proof Arguments 28 Proof of Convergence 134 Proof Technique 96 Proper Class 133 pth Root of Unity 42 pth Roots of Unity 44

Q Quadratic 84 Quadratic Energy 91 Quadratic Energy Function 22, 31, 34, 92 Quadratic Form 23, 31, 56, 58, 86, 91, 92, 95 Quadratic Form over the Hypercube 80 Quadratic Forms 12, 19 Quadratic Objective Function 57, 91, 92 Quadratic/Higher Degree Energy Function 10 Quarter Plane Causal Distributed Dynamical Systems 75 Quarter Plane Causal Model 75 Quarter Plane Causality 63, 64, 65, 70 Quarter-plane Causality, Half-plane Causality 62, 82 Quaternion Based Neural Networks 135

R Random Field 72 Random Field Models 71 Random Process 72 Random Variable 72 Rational Functions 63 Real Anti-symmetric One 97 Real Connection Matrix 102 Real Mode 100 Real Numbers 120 Real Part 97, 98, 99 Real Symmetric 97

Index

153

Real Valued Associative Memories 130 Real Valued Neural Networks 117 Real Valued Neuron 118 Real/Complex Valued Sequences 130 Realistic Model 117 Realizable 83 Received Tensor 43, 53 Received Tensor Word 50 Received Word 45 Recursion 124 Redundancy 4 Representation 38, 82 Reproduction 138 Response Determination 70 Ring 120 Robots 3 Roesser’s Model 64

S Samples 131 Scalar Synaptic Weight 117 Second Order Models 64 Separable Filters 64 Separating Line/Hyper Plane 132 Serial Mode 14, 20, 21, 22, 30, 96, 97, 99, 103, 104 Shannon 107 Sigmoid Function 134 Sign Structure 58 Signal Design 122 Signal Design Problem 24 Signed Integral Equation 123 Signum Function 105, 118, 120 Signum or Sigmoid or Hyperbolic Tangent 133 Simulated Annealing 27 Single Input, Single Output 81, 113 Single Input, Single Output Linear Time Invariant 123 Single Layer 117, 121, 133 Single Layer Neural Network 81 Single Layer of Perceptrons 118 Single Layer Perceptron 117 Single Synaptic Weight 117 Single/Multi-Layer Continuous Time Neural Networks 125

SISO Discrete Time, Linear Time Invariant Systems 81 Solution of the Difference Equation 85 Space 32 Space Representations 71 Special Sets 55 Species 6 Species of Living Systems 138 Spectral Representation Theorem 72 Spectral/Cholesky Type Decomposition 57 Sphere Packing Problem 45 Stability 62 Stable 14, 30 Stable State 16, 20, 21, 22, 30, 40, 58, 92, 96, 97, 102, 104, 113, 115 Stable State of a Continuous Time 89 Stable State of a Multidimensional Hopfield Neura 86 Stable States 17, 18, 19, 22, 80, 81, 91, 92, 108, 110, 113, 114 Stable States (Stable Functions) 5 Stable States of a Hopfield Network 92 Stable States of a Hopfield Neural Network 81 Stable States of a Multidimensional Hopfield Neural Network 82 Stable States of a One Dimensional Neural Network 81 Stable States of Multidimensional Neural (Generalized) Network 91 Stable States of Neural Network 81 Standard Theorems 55 State 13, 31, 67, 95 State Coupling Tensor 67, 68, 82 State Equations 88 State Estimation 73 State Evolution 30 State, Input 72 State of a Neuron 13 State of a Node 20 State of Neuron. 15 State of Node 30 State of the Dynamical System 68, 82 State of the Network 14, 20, 30 State of the Node 16 State Response 109 State Space 22, 27, 30, 63, 74

154

State Space Description 61, 67, 82, 107, 108 State Space Description of a Dynamical System 75 State Space Model 64 State Space Representation 58, 59, 61, 63, 65, 66, 69, 73, 76, 82, 91 State Space Representation Through Tensors 70 State Space Representations 72, 76 State Space Structure 69 State Tensor 13 State Transition Tensor 71, 72, 88 State Transitions 74 States of Neural Networks. 80 Static 63 Static Optimization 23, 29, 54, 55 Static Optimization Problems 53 Static Systems 65, 70 Stochastic Control Theory 73, 76 Stochastic Dynamic Programming 73 Stochastic Linear Systems 71 Stochastic Models 72 Stochastic Processes 71 Stochastic Tensor 37 Storage of Data 81 Stress Tensor 67 Structure of Optimal Control 110 Structure of the local Optimum 86 Structured Markov Random Field 71, 72, 73 Structured Markov Random Fields 71 Sub-spaces 34 Subsets of Multidimensional Lattice 19, 28 Subsets of the Lattice 51 Successive Approximation Procedure 123 Successive Approximation Scheme 123 Successive Input Functions 119, 134 Supervised Learning 135 Supervised Learning in a Function Space 135 Supervised Training 117 Switching/Logic Functions 19 Symmetric 99 Symmetric Matrix 11, 20, 58, 95 Symmetric Tensor 10, 18, 23 Synapse 117, 121 Synapses 117, 118, 126, 133 Synaptic Contribution 15 Synaptic Weight 134, 135 Synaptic Weight Function 121 Synaptic Weight Functions 118, 121, 125, 133, 134, 135

Multidimensional Neural Networks: Unified Theory

Synaptic Weight Matrix 56, 124 Synaptic Weight Sequence Values 122 Synaptic Weights 13, 15, 20, 25, 97, 117, 118, 133 Syndrome 41 Synthesis 61, 62 System 61 System Dynamics 75, 83, 87, 124 System Theorists 63, 68 System Theory 66, 107 System Theory Approach 107 Systematic Form 29 Systems 67, 139 Systems Function 61

T Target Output 119, 122, 134 Tensor 11, 29, 32, 68 Tensor Algebra, 49, 76 Tensor Algebra Concepts 35 Tensor Analysis 66 Tensor Based 29 Tensor Based Difference/Differential Equations 76 Tensor Based Multivariate Polynomials 28 Tensor Based State Space Representation 63 Tensor Field 74 Tensor Functions 74 Tensor Geometric Form 72 Tensor Inner Product (Outer Product) 35 Tensor Linear Operator 34, 35, 62, 63, 65, 69, 76 Tensor Linear Operators 62, 71, 76 Tensor Linear Space 36 Tensor Linear Spaces 59, 69, 70 Tensor of Partial Derivatives 76 Tensor of Probabilities of The States 71 Tensor Product 12, 48 Tensor Products 46, 73, 76, 125, 135 Tensor Products, Matrix Products 47 Tensor Spaces 28, 34 Tensor State Space 82 Tensor State Space Description 69, 70 Tensor State Space Representation 62, 63, 66, 67, 68, 71, 72, 73, 76, 80, 82, 92 Tensor State Space Representations 68 Tensor-tensor Products 76 Tensor-tensor Variables 70 Tensors 10, 12, 38, 58, 62 Terminal State 84

Index

Test Patterns 133 The Common Thread of Neural Networks 91 The Nervous System 79 The Real and Imaginary Parts 98 The Time Varying Linear System 85 Theorems 28 Theoretical Computer Science 56 Theory of Error Control Codes 45 Theory of Error Correcting Codes 39 Theory of Multidimensional Neural Networks 28 Three Dimensional Array 125 Three Dimensional Scenes 129 Three/Multidimensional Array 135 Threshold 13 Threshold Value 29 Thresholds 21, 29, 130, 135 Time Invariant Linear System 86 Time Series Model 73 Time Varying Synaptic Weight Matrix 123 Time Varying Systems 66 Time-invariant Systems 66 Topography 27 Topological 34 Topological Features 138 Total Output Energy 81, 108 Trainability of Intelligence 138 Training 138 Training Example 119, 122 Training Phase 117 Training Samples 117, 133 Transfer Function 107 Transformed Samples 132 Transitions 19 Transmitted Tensor 43 TSSR 73 Two Dimensional Distributed System 75 Two Dimensional Euclidean Space 132 Two Dimensional Filter Theory 63 Two Dimensional Neural Networks 10 Two Dimensional Signal Processing 75 Two Dimensional State Space Models 63 Two Dimensional System Theory 75 Two Possible States 13 Two/Multidimensional Logic Circuits 9 Two/Multidimensional Arrays 9 Two/Multidimensional Neural Networks. 11 Two/Multidimensional System Theory 62, 76 Two/Multidimensional Systems 63

155

Type of Neighbourhood of States 19

U Unbounded Lattice 24 Undirected Graph 57, 95 Uni-variate Scalar Polynomial 52 Unification 80, 90, 92 Unification of Control 91 Unified 81 Unified Theory 6, 92 Unified Theory of Control, Communication and Computation 80, 126, 137 Uniform Convergence 134 Univariate 51 Universe 1 Unknown Tensor X 53 Updating of the Function 99

V Various Recurrent Networks 120 Vector Space 120, 121 Vector-matrix Variables 70 Vector/Matrix Products 76 Version 124

W Weight Matrix 95, 97 Weight of a Cut 31 Weighted and Undirected Non-planar Graph 30 Weighted Contribution 135 Weighted Undirected Connectionist Structure 29 Weights 119 Weights are Complex 103 White As Well As Colored 73 White Noise 71 Wiener 107, 137 Wiener and Kalman Filters 73 Wyner 81, 122

Z Zero Mean Tensors 72 Zurada 105