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\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
\begin{equation} \label{equation}
    \frac{dZ} {dt} = TMZ-ZZ^*MZ,
\end{equation}
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
 \begin{equation} \label{E:MainPropose}
 \begin{gathered}
 \dot{Z}=  M Z   -  Z  Z^* M  Z \\
 Z(0)=Z_0.
\end{gathered}
\end{equation}
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
\begin{equation}   \label{E:VDot=MV+VM^*...}
\begin{gathered}
\dot{V}= MV   +
            VM^*   -  VMV   -  VM^*V \\
        V(0)=Z_0 Z_0^*.
\end{gathered}
\end{equation}
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
\begin{equation}\label{E:Pdotdefinitionwi}
\begin{gathered}
 \dot{P}=  -\frac{1}{2}(M-M^*)(V(t)-I)  P\\
     P(0)=P_0.
\end{gathered}
\end{equation}
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
    \begin{equation}  \label{E:MainProposed1}
 \begin{gathered}
\dot{Z}=     M Z   -  Z  Z^* M  Z \\
Z(0)=Z_0,
\end{gathered}
    \end{equation}
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
\begin{equation} \label{cox_adams}
  \begin{gathered}
\frac{dZ} {dt} = TMZ-ZZ^*MZ\\
Z(0)=Z_0
\end{gathered}
\end{equation}
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
    \begin{equation} \label{dotW}
 \dot{W}= S W   - W  W^* S  W \quad\text{and}\quad    W(0)=W_0,
 \end{equation}
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
    \begin{equation}  \label{E:NewMainProposed1}
\begin{gathered}
\dot{Z}=TM Z   -  Z  Z^* M  Z \\
Z(0)=Z_0,
\end{gathered}
    \end{equation}
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}

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\end{document}