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\begin{document}
\begin{LARGE}

%%%%%%%%%%  Title  %%%%%%%%%%
\clearpage
\begin{center}
\begin{Huge}
\noindent
{\bf Error Correcting Capability \\
     of a Morphological Associative Memory}\\
\end{Huge}
\vspace{1in}
{\bf
Akira Imada\\
Dokuz Eylul University\\
Izmir, Turkey
}
\end{center}

%%%%%%%%%%  To be more specific  %%%%%%%%%%
\clearpage
\noindent
%%$\Box${\bf To be more specific:}\\

\begin{itemize}
\item Traditional Associative Memory
   \begin{itemize}
   \item[] $p < 0.138 N$ $\cdots$ Hebb's rule (1982)
   \item[] $p < 2 N$     $\cdots$ Gardner (1988)
   \end{itemize}
\item[]
\item Morphological Associative Memory
   \begin{itemize}
   \item[] $p < 2^N$
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{itemize}
\item {\bf Infinitely large storage capacity!}
\end{itemize}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=fp-storage.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}
\begin{itemize}
\item {\bf But how about the size of basin-of-attraction?}
\end{itemize}

%%%%%%%%%%  Associative memory as a dynamical system  %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=what-is-am.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}

%%%%%%%%%%  Category of trajectory  %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=trajectory.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}

%%%%%%%%%%  Realization  %%%%%%%%%%
\clearpage
\noindent
$\Box${\bf The Associative Memory can be realized with:}\\
\begin{itemize}
\item fully-connected neural network model;
\item immune system model;
\item etc.
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=fig-nn.eps,width=16cm,height=7cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf To store patterns}\\
\begin{displaymath}
(\xi_1^\mu, \xi_2^\mu, \cdots, \xi_N^\mu),
 \hspace{.3in}\mu=1, 2, \cdots, p
\end{displaymath}


%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Hopfield Model}\\

\indent
$\bullet$ Hebbian weights:

\begin{displaymath}
w_{ij} = \frac{1}{N} \sum _{ \nu = 1}^{p} 
{{ \xi }_{i}^{ \nu }{ \xi }_{j}^{ \nu }}
\hspace{.1in}(i \ne j), \hspace{.1in} w_{ii} = 0
\end{displaymath}

$\bullet$ Rule for update:
\begin{displaymath}
s_{i}\hspace{.02in} (t+1) = sgn \hspace{.02in} 
\left( \hspace{.04in} \sum_{j \ne i}^N { w_{ij} s_{j}(t)} 
     \hspace{.04in} \right)
\end{displaymath}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Morphological Model}\\

\indent
$\bullet$ Weights:

\begin{displaymath}
w_{ij} = \mbox{min}_{\nu = 1}^{p} ({\xi}_{i}^{\nu} - {\xi}_{j}^{\nu})
\end{displaymath}

\indent
$\bullet$ Rule for update
\begin{displaymath}
s_{i}\hspace{.02in} (t+1) = sgn \hspace{.02in} 
\left( \hspace{.04in} \mbox{max}_{j = 1}^N {( w_{ij} + s_{j}(t))} 
     \hspace{.04in} \right)
\end{displaymath}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf An assertion in Morphological Model:}\\
\begin{itemize}
\item ``It has an unlimited storage capacity.''
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Recovery ratio:}\\
\begin{itemize}
\item Average number of bits of converged patterns
   \begin{itemize}
   \item[] \hspace{2in}that are unchanged from the input.
   \end{itemize}
\end{itemize}

%%%%%%%%%% Again
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=fp-storage.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Error Correcting Capability:}\\

\noindent

\begin{description}
\item[$\bullet$] A noisy input of a stored pattern converges to
          the perfect version of the pattern,
          if the number of noisy bits is within the basin of attraction.\\
\end{description}

\begin{itemize}
\item[--] Hopfield Model:
   \begin{itemize}
   \item[$\star$] in  several steps; or 
   \item[$\star$] sometimes via a long long tour.
   \end{itemize}
\item[--] Morphological Model:
   \begin{itemize}
   \item[$\star$] in one step for any input pattern.
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Errors in the Morphological Model:}\\

\begin{itemize}
\item erosive error
   \begin{itemize}
   \item[] that changes some of $+1$ of the input pattern into $-1$, 
   \end{itemize}
\item dilative error 
   \begin{itemize}
   \item[] that changes some of $-1$ into $+1$.\\
   \end{itemize}
\end{itemize}

\noindent
are processed by two different modules, 
though we focus only on erosive errors here.


%%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=susner-noisy.eps,width=16cm,height=8cm}
   \epsfile{file=hopf-noisy.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=sus-storage-cap.eps,width=16cm,height=8cm}
   \epsfile{file=hopf-storage-cap.eps,width=16cm,height=8cm}
 \end{center}
\end{figure}

%%%%%%%%%% Conclusion %%%%%%%%%%
\clearpage
\noindent
{\bf $\Box$ Conclusions:}\\
\begin{itemize}
\item Storage capacity of morphological associative memory is very large.
\item But the basin size is suffered from the large storage 
      more or less as the Hopfield Model.
\item Can we apply it to real-world problems?
\end{itemize}

\end{LARGE}
\end{document}


