\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. {\em Electronic Journal of Differential Equations}, Conf. 19 (2010), pp. 31--36.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \setcounter{page}{31} \title[\hfilneg EJDE-2010/Conf/19/\hfil Models of learning] {Models of learning and the polar decomposition of bounded linear operators} \author[F. Botelho, A. Davis\hfil EJDE/Conf/19 \hfilneg] {Fernanda Botelho, Annita Davis} % in alphabetical order \address{Fernanda Botelho \newline The University of Memphis\\ Mathematical Sciences Dept.\\ Dunn Hall 373, 3721 Norriswood St.\\ Memphis, TN 38152-3240, USA} \email{mbotelho@memphis.edu} \address{Annita Davis \newline The University of Memphis\\ Mathematical Sciences Dept.\\ Dunn Hall 373, 3721 Norriswood St.\\ Memphis, TN 38152-3240, USA} \email{adavis2@memphis.edu} \thanks{Published September 25, 2010.} \subjclass[2000]{34G20, 47J25} \keywords{Nonlinear systems; learning models; \hfill\break\indent polar decomposition of operators} \begin{abstract} We study systems of differential equations in $\mathcal{B}(\mathcal{H})$, the space of all bounded linear operators on a separable complex Hilbert space $\mathcal{H}$ equipped with the operator norm. These systems are infinite dimensional generalizations of mathematical models of learning. We use the polar decomposition of operators to find an explicit form for solutions. We also discuss the standard questions of existence and uniqueness of local and global solutions, as well as their long-term behavior. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Learning models are devices that reproduce features of the human's ability to interact with the environment. Such models are typically implemented in a feedforward neural network consisting of a finite number of interconnected units, designated neurons. Each neuron has the capability of receiving, combining and processing quantifiable information. The interconnection among neurons is represented by a net of channels through which information flows. This translates the activity of a brain at the synaptic level. It is natural to predict that information changes in this system. This is represented by the action of multiplicative factors, designated connecting weights. A mathematical model of learning should encompass an algebraic interpretation of this phenomenon, in general, given as a system of differential equations. The stability of a learning model is of crucial importance since it provides information on how the device emerges after exposed to a set of initial conditions. If stability occurs, the result yields a set of weight values that represents learning after the exposure to an initial stimulus. In this paper we address this aspect of learning theory, often referred in literature as unsupervised learning". Several researchers have proposed systems that perform unsupervised learning, see \cite{Amari1, Haykin, Hertz}. In \cite{Oja} Oja introduced a learning model that behaves as an information filter with the capability to adapt from an internal assignment of initial weights. This approach uses the principal component analyzer statistical method to perform a selection of relevant information. More recently, Adams \cite{Adams} proposed a generalization of Oja's model by incorporating a probabilistic parameter. Such a parameter captures the possible creation of temporary synapses or channels in an active network. We refer the reader to \cite{Botelho2} for a detailed interpretation of the Cox-Adams model for a network with $n$ input neurons and a single output neuron. This model is given by the system of differential equations: $\frac{dW}{dt}=T CW- WW^TCW$ with $W$ a column of connecting weights and $C$ is a symmetric matrix. Each entry value in $C$ is equal to the expected input correlation between two neurons. The matrix $T$ is a non-singular tri-diagonal matrix $[ t_{ij}]_{i,j}$ given by $t_{i j}=\begin{cases} 1-\varepsilon &\text{if }i=j\\ \varepsilon/2 &\text{if } |i-j|=1\\ 0 &\text{otherwise}. \end{cases}$ This matrix translates the synaptic formation according to some probability $\epsilon$. In this paper we consider a generalization of this system to an infinite dimensional setting. This better reflects the complexity of real systems where continuous activity occurs. Our new setting is the Banach space $\mathcal{B}(\mathcal{H})$ of bounded operators on a separable Hilbert space . We consider the system $$\label{equation} \frac{dZ} {dt} = TMZ-ZZ^*MZ,$$ with $T$ representing an invertible, positive, self-adjoint operator on $\mathcal{H}$ and M a self-adjoint operator on $\mathcal{H}$. Particularly interesting examples are the tridiagonal self-adjoint operators, see \cite{Dombrowski1, Dombrowski2, Duren1, Duren2}. The operator valued, time dependent $Z$ now represents the continuous change of connecting weights according to the rule described in equation \eqref{equation}. We present a scheme that explicitly solves system \eqref{equation}. First a natural change of variables reduces \eqref{equation} to a static system where no synaptic formation occurs. However, the probabilistic effect transfers to the input correlation operator M. System \eqref{equation} reduces to an Oja type model. We follow a strategy employed in \cite{Botelho1}. The main tool is the polar decomposition of operators that allows us to derive a scalar system and a polar system associated with the original system. Both systems are solved explicitly. These two solutions combined define the local solution for the original system, given certain mild constrains on the initial conditions. The explicit form for local solutions is now used to derive the existence of global solutions and for the stability analysis. \section{Background Results} In this section we summarize the techniques used in \cite{Botelho1} to solve the generalization of Oja-Karhunen's model on a separable Hilbert space. We recall that the generalized Oja-Karhunen model is given as follows $$\label{E:MainPropose} \begin{gathered} \dot{Z}= M Z - Z Z^* M Z \\ Z(0)=Z_0. \end{gathered}$$ The time dependent variable $Z$ has values in $\mathcal{B(H)}$. The operator $Z^*$ is the adjoint of $Z$ and $M$ is a normal operator on $\mathcal{H}$. Classical fixed point theorems allow us to assure the local existence and uniqueness of solutions for system \eqref{E:MainPropose}, see \cite[p. 405]{Hartman}. \begin{theorem}[\cite{Botelho1}] \label{EUS} Let $Z_0$ be a bounded operator in $\mathcal{B}(\mathcal{H})$. If $F:\mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})$ is a Lipschitz function then there exists a positive number $\varepsilon$ and a unique differentiable map $Z:(-\varepsilon, \varepsilon) \to \mathcal{B(H)}$ such that $\dot{Z} = F(Z)$ and $Z(0) = Z_0$. \end{theorem} It is straightforward to check that $Z \to MZ-ZZ^*MZ$ is a Lipschitz function and hence local existence and uniqueness of solutions of \eqref{E:MainPropose} follow from Theorem \ref{EUS}. The technique employed in the solution scheme uses the well known polar decomposition of bounded operators, see Ringrose \cite{Ringrose}. A bounded operator $Z$ can be written as the product of a partial isometry P and a hermitian operator $\sqrt{ZZ^*}$. The operator $\sqrt{ZZ^*}$ represents the unique positive square root of $ZZ^*$ and the operator $P$ satisfies the equation $PP^*P=P.$ The bounded operator $Z$ is decomposed as follows: $Z= \sqrt{ZZ^*} P$. This decomposition is unique and is called the polar decomposition of the operator $Z$. In \cite{Botelho1} we applied the polar decomposition to construct two new systems associated with \eqref{E:MainPropose}. The solutions of these systems are the scalar and the polar components of the solution of the original system. For $t \in (-\varepsilon, \varepsilon)$ we denote by $Z(t)$ a local solution of \eqref{E:MainPropose} and we set $V(t)= Z(t)Z(t)^*$. It is a straightforward calculation to verify that $V(t)$ is a local solution of the system $$\label{E:VDot=MV+VM^*...} \begin{gathered} \dot{V}= MV + VM^* - VMV - VM^*V \\ V(0)=Z_0 Z_0^*. \end{gathered}$$ If $M$ and $Z_0$ commute, Fuglede-Putman Theorem (cf. Furuta \cite[p. 67]{Furuta}) and Piccard's iterative method (Hartman \cite[pp. 62-66]{Hartman}) imply that the family $\{V(t)\}_{t \in (-\epsilon, \epsilon)}$ defines a path of positive operators that commute with $M$ and $M^*$. Furthermore $\{V(t)\}_{t \in (-\epsilon, \epsilon)}$ is a family of commuting operators. Thus system \eqref{E:VDot=MV+VM^*...} can be written as $$\dot{V}= (M+M^*)( V - V^2), \quad \text{with } V(0)=V_0.$$ Since $Z_0$ is an invertible operator in $\mathcal{H}$, for some $\varepsilon > 0$, $V(t)$ is also an invertible operator for $t \in (\varepsilon, \varepsilon)$, cf \cite{Douglas}. We have that $\frac d{dt}(V^{-1})=-V^{-2}\dot{V}.$ Using the commutativity of $M$ and $V$, we found that $\frac d{dt}(V^{-1})=(M+M^*) - (M+M^*)V^{-1},$ which is a first-order linear differential equation, see \cite{Abell}. A generalization of standard techniques of integrating factors, appropriately generalized to infinite dimensions (see \cite{Botelho1} pg 101), imply that $V(t)=[I+(V_0^{-1}-I) exp (-(M+M^*)t)]^{-1}$ for $t \in (-\epsilon, \epsilon)$. We derive the polar system associated with \eqref{E:MainPropose}. This is a first order non autonomous linear differential equation. For simplicity of notation we set $V^{1/2}= \sqrt{ZZ^*}$. Using the commutativity of $V$ and $M$, we obtain $$\label{E:Pdotdefinitionwi} \begin{gathered} \dot{P}= -\frac{1}{2}(M-M^*)(V(t)-I) P\\ P(0)=P_0. \end{gathered}$$ Properties of exponential operator-valued functions imply that $$P(t)=\exp\Big(\int_0^t -\frac{1}{2}(M-M^*)(V(\xi)-I) d\xi\Big) P_0 \quad \text{for } | t|<\varepsilon)$$ is a solution of the polar system \eqref{E:Pdotdefinitionwi}. These considerations are now summarized in the following theorem. \begin{theorem}[\cite{Botelho1}] \label{maintheorem} If $Z_0$ is invertible and commutes with the normal operator $M,$ then there exist $\varepsilon >0$ and a unique differentiable mapping $Z: (-\varepsilon, \varepsilon) \to \mathcal{B}(\mathcal{H})$ such that $$\label{E:MainProposed1} \begin{gathered} \dot{Z}= M Z - Z Z^* M Z \\ Z(0)=Z_0, \end{gathered}$$ if and only if $Z(t)=V(t)^{1/2}P(t)$, $V(t)=[I+(V_0^{-1}-I) \exp (-(M+M^*)t)]^{-1},$ and $P(t)=\exp\Big(\int\limits_{0}^{t}-\frac{1}{2}(M-M^*)(V(\xi)-I) d\xi\Big)P_0.$ \end{theorem} \section{General Solution for the Cox-Adams Learning Model} We recall that the Cox-Adams learning model is $$\label{cox_adams} \begin{gathered} \frac{dZ} {dt} = TMZ-ZZ^*MZ\\ Z(0)=Z_0 \end{gathered}$$ with $T$ representing an invertible, positive, self-adjoint operator, and $M$ self-adjoint on $\mathcal{H}$. Theorem \ref{EUS} implies the local existence and uniqueness of solutions. Since $T$ is positive and invertible, we rewrite equation \eqref{cox_adams} as follows $\frac{dZ} {dt} = (\sqrt{T}\sqrt{T}) M \big(\sqrt{T}(\sqrt{T})^{-1}\big) Z -ZZ^* \big((\sqrt{T})^{-1}\sqrt{T}\big) MZ,$ equivalently, $%\label{nform} (\sqrt{T})^{-1}\frac{dZ} {dt} = \sqrt{T} M\sqrt{T}(\sqrt{T})^{-1}Z -(\sqrt{T})^{-1}ZZ^*(\sqrt{T})^{-1} \sqrt{T}MZ.$ We set $W = (\sqrt{T})^{-1}Z$ and $S=\sqrt{T} M\sqrt{T}$. We observe that $S$ is a hermitian operator. Then system \eqref{cox_adams} becomes \begin{gather*} \dot{W}= SW -WW^*SW \\ W(0)=W_0, \end{gather*} where $W_0 = (\sqrt{T})^{-1} Z_0$. \begin{proposition} \label{L:TepsilonFinal} If $W_0$ is invertible and commutes with the hermitian operator $S$, then there exist $\varepsilon >0$ and a unique differentiable mapping $W: (-\varepsilon, \varepsilon) \to \mathcal{B}(\mathcal{H})$ such that $$\label{dotW} \dot{W}= S W - W W^* S W \quad\text{and}\quad W(0)=W_0,$$ if and only if $W(t)=V(t)^{1/2}P(t)$, with \begin{gather} \label{VwithW} V(t)=[I+(V_0^{-1}-I) \exp(-2St) ]^{-1}\\ \label{PwithS} P(t)=P_0. \end{gather} \end{proposition} Since the operator $S$ is hermitian $(S=S^*)$, the proof of the above lemma follows from Theorem \ref{maintheorem} \begin{theorem} \label{Newmaintheorem} Let $T$ be an invertible, positive, self-adjoint operator and $M$ a hermitian operator. If $Z_0$ is invertible and commutes with $M,$ then there exist $\varepsilon >0$ and a unique differentiable mapping $Z: (-\varepsilon, \varepsilon) \to \mathcal{B}(\mathcal{H})$ such that $$\label{E:NewMainProposed1} \begin{gathered} \dot{Z}=TM Z - Z Z^* M Z \\ Z(0)=Z_0, \end{gathered}$$ if and only if $Z(t)=(TV(t))^{1/2}P(t)$, $V(t)=\big[I+\big(\sqrt{T}(Z_0Z_0^*)^{-1}\sqrt{T}-I\big) \exp(-2\sqrt{T} M \sqrt{T} t)\big]^{-1}$ and $P(t)= P_0$. \end{theorem} The proof of the above theorem follows from Proposition \ref{L:TepsilonFinal}. The following lemma is used in the stability analysis of the Cox-Adams model. \begin{lemma}[\cite{Botelho1}] \label{L:Zexists1} If $Z_0$ is an invertible operator in $\mathcal{B}(\mathcal{H})$, $M$ is a normal operator that commutes with $Z_0$, $\|(Z_0 Z_0^*)^{-1}-I \|<1$, and the spectrum of $M$ is strictly positive, then there exists $\epsilon >0$ so that $I+[(Z_0 Z_0^*)^{-1}-I] \exp(-(M+M^*) t)$ is invertible on the interval $(-\varepsilon, \infty)$ and $\lim_{t \to \infty} [I+((Z_0 Z_0^*)^{-1}-I) \exp (-(M+M^*) t ) ]=I.$ \end{lemma} As a result we have the following corollary. \begin{corollary}\label{L:NewZexists1} Let $T$ be an invertible, positive, self-adjoint operator. If $Z_0$ is an invertible operator in $\mathcal{B}(\mathcal{H})$, $M$ is a self-adjoint operator that commutes with $Z_0$, $\|\sqrt{T}(Z_0 Z_0^*)^{-1}\sqrt{T}-I \|<1$, and the spectrum of $M$ is strictly positive, then there exists $\epsilon >0$ so that $I+[\sqrt{T}(Z_0 Z_0^*)^{-1}\sqrt{T}-I] \exp \big(-2\sqrt{T}M\sqrt{T}) t \big)$ is invertible on the interval $(-\varepsilon, \infty)$ and $\lim_{t \to \infty} \big[I+[\sqrt{T}(Z_0 Z_0^*)^{-1}\sqrt{T}-I] exp\big(-2\sqrt{T} M\sqrt{T} t \big) \big]=I.$ \end{corollary} \begin{remark} \label{Newstability} \rm We observe that, under the assumptions in Corollary \ref{L:NewZexists1}, we have $\lim_{t \to \infty}Z(t) = P_0.$ This provides a filtering procedure that selects the polar component of the initial condition. \end{remark} \begin{thebibliography}{99} \bibitem{Abell} M. Abell, J. Braselton; \emph{Modern Differential Equations}, 2nd ed., Harcourt College Publishers, (2001), 36-421, 197-205. \bibitem{Adams} P. 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