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\begin{document}
\begin{LARGE}

%%%%%%%%%%  3. Asociative memory as a dynamical system  %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=what-is-am.eps,width=21cm,height=14cm}
 \end{center}
\end{figure}

%%%%%%%%%%  4.  cotegory of trajectory  %%%%%%%%%%
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 \begin{center}
   \epsfile{file=trajectory.eps,width=24cm,height=14cm}
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%%%%%%%%%%  1. title  %%%%%%%%%%
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\begin{center}
\begin{Huge}
\noindent
{\bf A Genetic Algorithm\\using an Entropy based Selection Scheme}\\
\end{Huge}
\vspace{1in}
{\bf
Akira Imada\\
Nara Institute of Science and Technology\\
(akira-i@is.aist-nara.ac.jp)
}
\end{center}

%%%%%%%%%%  2. Introduction  %%%%%%%%%%
\clearpage
{\bf $\Box$ Introduction:}\\

\begin{itemize}
\item We apply 
   \begin{itemize}
   \item ECs to neural network model of Associative Memory.
   \end{itemize}
\item We study
   \begin{itemize}
   \item
 basic behaviors of the model under simple evolutionary processes.
   \end{itemize}
\item Here,
\begin{itemize}
   \item we use a selection scheme based on entropy of the population. \\(Lee and Han, 1997)
   \end{itemize}
\end{itemize}

%%%%%%%%%%  7. example of weights for N=3 %%%%%%%%%%
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\begin{figure}[tbph]
 \begin{center}
   \epsfile{file=three-neuron-sim.eps,width=21cm,height=14cm}
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\end{figure}

%%%%%%%%%%  8. concept of fitness landscape %%%%%%%%%%
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\begin{center}
{\bf Fitness Landscapes on Weight Space}\\

\vspace{.6in}

$\bullet$ all the possible configurations of $w_{ij}$ of a network with $N$\\

$\Downarrow$ \\

$\bullet$ an $N^2$-dimensional Euclidean space \\

$\Downarrow$ \\

$\bullet$ a specific configuration is represented as a point in the space\\

$\Downarrow$ \\

$\bullet$ ``fitness landscape'' can be defined on the space

$\Downarrow$ \\

$\bullet$ where we can imagine the optima as peaks

\end{center}

%%%%%%%%%%  5. Realization  %%%%%%%%%%
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{\bf
\noindent
The Associative Memory can be realized with:\\
\begin{itemize}
%%\item neural network model \\ \hspace*{1.9in}(leyered/recurrent/self-organized)\\
\item fully-connected neural network model \\ \hspace*{4.9in}(Hopfield model)\\
\item immune system model\\
\item etc
\end{itemize}
}

%%%%%%%%%%  6. Schematic diagram of hopfield model for N=3 %%%%%%%%%%
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\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=fig-nn.eps,width=21cm,height=10cm}
 \end{center}
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%%%%%%%%%% 11. Hamming vs time -- 3  %%%%%%%%%%
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\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=trajectory-1000.eps,width=7cm,height=5cm}
   \epsfile{file=trajectory-2200.eps,width=7cm,height=5cm}
   \epsfile{file=trajectory-final.eps,width=7cm,height=5cm}
 \end{center}
\end{figure}

%%%%%%%%%%  12. string space vs weight space %%%%%%%%%% 
\clearpage
\noindent
{\bf $\Box$ Weight Space and String Space:}\\

\begin{itemize}
\item weight space
   \begin{itemize}
   \item a given set of patterns to be memorized: fixed
   \item $N^2$-dimensional; real-valued genes
   \end{itemize}
\item[]
\item string space
   \begin{itemize}
   \item a configuration of weights: fixed
   \item $N$-dimensional; binary genes
   \end{itemize}
\end{itemize}

%%%%%%%%%% 9. Hamming vs time -- 1 %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=trajectory-0154.eps,width=10cm,height=7cm}
   \epsfile{file=trajectory-0200.eps,width=10cm,height=7cm}
 \end{center}
\end{figure}

%%%%%%%%%% 10. Hamming vs time -- 2  %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=trajectory-0210.eps,width=7cm,height=5cm}
   \epsfile{file=trajectory-0240.eps,width=7cm,height=5cm}
   \epsfile{file=trajectory-0300.eps,width=7cm,height=5cm}
 \end{center}
\end{figure}

%%%%%%%%%%  15. entropic sampling  %%%%%%%%%% 
\clearpage
\begin{center}
{\bf Entropic Sampling:}\\

\vspace{.6in}

$\bullet$ Fitness level is devided into 21 classes (from row to high).\\

$\Downarrow$ \\

$\bullet$ The last class is only for perfect fitness.\\

$\Downarrow$ \\

$\bullet$ The fewer the indivisuals are, the higher the probability to be selected.\\

\vspace{.2in}

({\bf cf.} fitness proportionate selection.)\\

\vspace{.2in}

$\Downarrow$ \\

$\bullet$ Can we locate multiple peaks at a run?

\end{center}

%%%%%%%%%% 16. fitness histogram %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=histogram-0.eps,width=4.5cm,height=12cm}
   \epsfile{file=histogram-1.eps,width=4.5cm,height=12cm}
   \epsfile{file=histogram-2.eps,width=4.5cm,height=12cm}
   \epsfile{file=histogram-3.eps,width=4.5cm,height=12cm}
   \epsfile{file=histogram-4.eps,width=4.5cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%  13. hill-climbing on string space  %%%%%%%%%% 
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=5100-rmhc.eps,width=24cm,height=14cm}
 \end{center}
\end{figure}

%%%%%%%%%%  14. sga on string space  %%%%%%%%%% 
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=fit-simple.eps,width=24cm,height=14cm}
 \end{center}
\end{figure}

%%%%%%%%%% 17. fitness under entropic sampling %%%%%%%%%%
\clearpage
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=fitness.eps,width=24cm,height=14cm}
 \end{center}
\end{figure}

%%%%%%%%%% 18. conclusion %%%%%%%%%%
\clearpage
{\bf $\Box$ Conclusions:}\\
\begin{itemize}
\item The ``entropic sampling'' works as well as simple GA.
   \begin{itemize}
   \item The more complicated landscape, the more effective?
   \item Can we locate multiple solutions simultaneously? 
   \end{itemize}
\item[]
\item These studies shed new light on the analysis of the Hopfield model of associative memory.
\end{itemize}

\end{LARGE}
\end{document}
