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\begin{document}
\begin{LARGE}

%%%%%%%%%%  Title  %%%%%%%%%%
\clearpage
\begin{center}
\begin{Huge}
\noindent
{\bf High-dimensional Weight Space.

\hspace*{1in}\\

\hspace*{1.5in}--- How can we Observe it?}\\
\end{Huge}
\vspace{1in}
{\bf
Akira Imada\\
\hspace*{1in}\\
Dokuz Eylul University\\
Izmir, Turkey
}
\end{center}


%%%%%%%%%% to be more specific %%%%%%%%%% 
\clearpage
\begin{center}
{\bf We Evolve Neural Networks.}\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\bf Search for an Appropriate Weight Configuration.}\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\bf Solutions are not usually unique.}\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\bf We need an information of Weight Space\\that is High-Dimensional.}
\end{center}


%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf NN \& GA}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=nn.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=fitness-ls-2d.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}


%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Search for a Needle in a Haystack.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=histogram.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}


%%%%%%%%%%
\clearpage
\begin{center}
{\bf Evolution\\
\hspace*{1in}\\
of\\
\hspace*{1in}\\
Associative Memory Model\\
\hspace*{1in}\\
using\\
\hspace*{1in}\\
Fully-connected Neural Network\\
\hspace*{1in}\\
by\\
\hspace*{1in}\\
Evolutionary Computations.}
\end{center}

%%%%%%%%%%  Content  %%%%%%%%%%
\clearpage
\noindent
$\Box${\bf Contents:}\\

\begin{itemize}
\item What is Associative Memory?
\item How we evolve it?\\
\item[] \underline{\hspace*{5in}} \\
\item Exploration on:\\
   \begin{itemize}
   \item[$\cdot$]High-D Fitness Landscape\\
      \begin{itemize}
      \item[] \hspace{.3in}(1) Random Hill-climbing.\\
      \end{itemize} 
   \item[$\cdot$]High-D Weight Space\\
      \begin{itemize}
      \item[] \hspace{.3in}(2) Principal Component Analysis (PCA);
      \item[] \hspace{.3in}(3) Kohonen's Topology-preserving Map;
      \item[] \hspace{.3in}(4) Sammon Mapping.
      \end{itemize} 
   \end{itemize}
\item[]\underline{\hspace*{5in}} \\
\item Concluding Remarks
\item Future Work
\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}

%%%%%%%%%%
\clearpage
\noindent
$\Box${\bf Associative Memory:}\\

\begin{itemize}
\item stores patterns:
   \begin{itemize}
   \item in a distributed way,
   \end{itemize}
\item recalls patterns:
   \begin{itemize}
   \item from noisy and/or partial input.
   \item[]\hspace*{.5in}(using attraction by attractors)\\
   \item[] \hspace*{1.5in}$\Downarrow$ i.e.\\
   \item {\it imperfect information} recalls {\it perfect information}
   \item[] \hspace*{.5in}with its capability depending on 
                         basin of attraction\\
   \end{itemize}
\end{itemize}

\noindent
We should note that there exists:
\begin{itemize}
\item trade-off between {\it storage capacity} and {\it basin size}.
\end{itemize}

%%%%%%%%%%  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
\begin{figure}[hbtp]
 \begin{center}
   \epsfile{file=three-neuron-sim-ex1.eps,width=16cm,height=5cm}
   \underline{\hspace*{6in}} \\
   \hspace*{1in}\\
   \epsfile{file=three-neuron-sim-ex3.eps,width=16cm,height=5cm}
   \underline{\hspace*{6in}} \\
   \hspace*{1in}\\
   \epsfile{file=three-neuron-sim-ex2.eps,width=16cm,height=5cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf $\Box$ State Transition}\\
\hspace*{1in}\\
\begin{displaymath}
s_i(t+1)= sgn \Biggl(\sum_j^N w_{ij}s_j(t)\Biggr)
\end{displaymath}
\hspace*{1in}\\
\hspace*{1in}\\
\noindent
$\Downarrow$ e.g.\\

\begin{eqnarray}
\nonumber
s_1(1) &=& sgn \Biggl( 
                  w_{11}\cdot s_1(0)+w_{12}\cdot s_2(0)+w_{13}\cdot s_3(0)
               \Biggr) \\
\nonumber
       &=& sgn \Biggl( 
                  0.14 \cdot 1 + (-0.34) \cdot 1 + (-0.52) \cdot 1
               \Biggr) \\
\nonumber
       &=& -1
\end{eqnarray}

\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf $\Box$ Hebbian Weights}\\
\hspace*{1in}\\
\begin{displaymath}
w_{ij}=\frac{1}{N}\sum_{\mu=1}^{p}\xi_i^\nu \xi_j^\nu, 
       \hspace{.2in} (i \ne j),
       \hspace{.2in} w_{ij}=0.
\end{displaymath}

\hspace*{1in}\\
\hspace*{1in}\\
\noindent
To store (1, 1, 1) in our previous toy example:

\begin{eqnarray}
\nonumber
\left(
\begin{array}{ccc}
 0    & 0.11 & 0.11 \\
 0.11 & 0    & 0.11 \\
 0.11 & 0.11 & 0    \\
\end{array}
\right)
\end{eqnarray}
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf
Thus we have multiple solutions in weight space.\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
A large network has an infinite number of \\
these solutions.\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
How can we obtain them exhaustively?\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
Evolutionary Computations.
}
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf To apply an EC to a problem} \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
A number of {\it candidate solutions} are produced\\ (mostly) at random \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
with each one represented as a string ({\it chromosome/individual})\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\it population}
\end{center}

%%%%%%%%%%
\clearpage
\begin{large}
\begin{eqnarray}
\nonumber
\left(
\begin{array}{rrr}
 0.14 & -0.34 & -0.52 \\
 0.62 &  0.27 & -0.47 \\
-0.22 & -0.04 &  0.77 \\
\end{array}
\right) = (0.14, -0.34, -0.52, 0.62 , 0.27 , -0.47, -0.22 , -0.04, 0.77)
\end{eqnarray}
\begin{eqnarray}
\nonumber
\left(
\begin{array}{rrr}
 0.82 & -0.39 & -0.45 \\
 0.38 & -0.24 &  0.70 \\
-0.58 &  0.46 & -0.88 \\
\end{array}
\right) = (0.82, -0.39, -0.45, 0.38 , -0.24 , 0.70, -0.58 , 0.46, -0.88)
\end{eqnarray}
\begin{center}
......\\
......\\
......\\
\end{center}
\begin{eqnarray}
\nonumber
\left(
\begin{array}{rrr}
-0.62 &  0.28 & -0.61 \\
 0.92 & -0.24 &  0.06 \\
-0.11 & -0.85 &  0.49 \\
\end{array}
\right) = (-0.62, 0.28, -0.61, 0.92 , -0.24 , 0.06, -0.11 , -0.85, 0.49)
\end{eqnarray}
\end{large}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf For an evolution to start} \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
we need \\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\it fitness} of each individual to the environment.\\
(how good is an individual?)\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
In our example:\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
 {\it the degree to how a weight configuration works 
     as an associative memory}

\hspace*{1in}\\
\hspace*{1in}\\
\underline{\hspace*{7in}} \\
\hspace*{1in}\\
\hspace*{1in}\\
At the start: \\
\hspace*{1in}\\
$\star$ they are low; \\
but \\
$\star$ different from each other!
\end{center}

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

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Evolution proceeds by:} \\

\begin{itemize}
\item Selection
\item Recombination
\item Mutation
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf How to observe an evolution?}\\

\begin{itemize}
\item[(1)] State trajectories.
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=trajectory-0154.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=trajectory-0200.eps,width=5.8cm,height=5.8cm}
 \end{center}
\end{figure}
%%%%%%%%%%

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=trajectory-0210.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=trajectory-0240.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=trajectory-0300.eps,width=5.8cm,height=5.8cm}
 \end{center}
\end{figure}

\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=trajectory-1000.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=trajectory-2200.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=trajectory-final.eps,width=5.8cm,height=5.8cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf How to observe an evolution?}\\

\begin{itemize}
\item[(2)] Fitness of the best individual in the population.
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=fit-r.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=fit-h.eps,width=5.8cm,height=5.8cm}
   \epsfile{file=fit-z.eps,width=5.8cm,height=5.8cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Storage Capacity}\\

\begin{itemize}
\item Storage Capacity under Hebbian Learning
   \begin{displaymath}
   p<0.138 N\\
  \end{displaymath}
\item[]
\item Absolute Limit of Storage (Gardner, 1988)
   \begin{displaymath}
   p<2N\\
   \end{displaymath}
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{itemize}
\item[$\Box$] {\bf Test of the diversity of the solutions\\
                   by two parameters during an evolution:}\\
   \begin{itemize}
   \item[(1)] Degree-of-Symmetry;\\
   \item[(2)] Rate-of-Zero.
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
{\bf Degree of Symmetry during Evolution.}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=symm-ep-97.eps,width=5.9cm,height=5.9cm}
   \epsfile{file=symm-icec-96.eps,width=5.9cm,height=5.9cm}
\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\
   \epsfile{file=symm-tainn-98.eps,width=5.9cm,height=5.9cm}
   \epsfile{file=symm-ieice-1.eps,width=5.9cm,height=5.9cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Other Interesting Topics}\\

\begin{itemize}
\item We can enhance a performance 
   \begin{itemize}
   \item[]by pruning some synapses adaptively.
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf Are there more direct ways\\ to observe the whole solutions?}\\
\hspace*{1in}\\
$\Downarrow$\\
\hspace*{1in}\\
{\bf Fitness Landscapes}
\end{center}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf The preliminary step:}

\begin{itemize}
\item Many different solutions by ECs.\\
\end{itemize}

\hspace*{1in}\\
\noindent
$\Box$ {\bf The next steps:}
\begin{itemize}
\item What does the Fitness Landscape look like?
   \begin{itemize}
   \item How does its geometry look?\\
   \end{itemize}
\item How about the Weight Space?
   \begin{itemize}
   \item The number and distribution of weight solutions?
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Fitness Landscape}\\

\begin{itemize}
\item The concept was first introduced 
      by Wright (1932) to study biological evolutionary processes. \\
\item Since then, it has been used not only 
      in evolutionary biology but also in chemistry, physics, 
      computer science and so on.
\end{itemize}

%%%%%%%%%%  Concept of Fitness Landscape %%%%%%%%%%
\clearpage
\begin{center}
{\bf Fitness Landscapes on Weight Space}\\
\vspace{.6in}
$\bullet$ 
all the possible configurations of $w_{ij}$ of a network\\with $N$ neurons\\
$\Downarrow$ \\
$\bullet$ 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}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf A Cross-section by a Hyperplane of the Hebbian Peak}
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=path-on-top-1.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}


%%%%%%%%%%
\clearpage
\begin{center}
{\bf Walkers\\
\hspace*{1in}\\
\hspace*{1in}\\
with its movement being defined by\\
\hspace*{1in}\\
\hspace*{1in}\\
Evolutionary Programming\\
\hspace*{1in}\\
\hspace*{1in}\\
walk down a hill\\
\hspace*{1in}\\
\hspace*{1in}\\
from the top of a peak\\
\hspace*{1in}\\
\hspace*{1in}\\
along a steepest path, hopefully,\\
\hspace*{1in}\\
\hspace*{1in}\\
revealing\\
\hspace*{1in}\\
\hspace*{1in}\\
the geometry of the landscape.
}
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf Evolutionary Programming}\\
\hspace*{1in}\\
{\bf employs}\\
\hspace*{1in}\\
{\bf only MUTATION}\\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
{\bf enables walkers to take}\\
\hspace*{1in}\\
{\bf small step sizes.}\\
\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\
{\bf CROSSOVER}\\
\hspace*{1in}\\
$\Downarrow$ \\
\hspace*{1in}\\
{\bf more like a jump than a walk.}
\end{center}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf A Typical Trace from Two Peaks.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=peak-shape.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Why shape of the peak?}\\
\begin{itemize}
\item We suspect that:\\
   \begin{itemize}
   \item[$\star$] the number of storage affects the peak shape;\\
   \item[$\star$] the peak shape affects the difficulty in search.\\
   \end{itemize}
\item We want to know:\\
   \begin{itemize}
   \item[$\star$] what is the fate of the local optima?\\
   \end{itemize}
\item etc.
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
{\bf $\Box$ Question 1.}\\

{\it Does the number of storage affect the peak shape?}\\
   \begin{itemize}
   \item[] We study this issue by observing {\it Hebbian peaks}\\
           in terms of:
      \begin{itemize}
      \item[]\hspace*{.2in} $\star$ width of the top;
      \item[]\hspace*{.2in} $\star$ steepness of the sidewall.
      \end{itemize}
   \end{itemize}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf Conjecture 1.}\\
\hspace*{1in}\\
\hspace*{1in}\\
Step sizes should be larger than a critical value,\\
\hspace*{1in}\\
\hspace*{1in}\\
otherwise
\hspace*{1in}\\
\hspace*{1in}\\
walkers remain at the top.\\
\hspace*{1in}\\
due to the basin of attraction or synaptic plasticity
\hspace*{1in}\\
\hspace*{1in}\\
(captured by an attractor)\\
\hspace*{1in}\\
\hspace*{1in}\\
  $\Downarrow$\\
\hspace*{1in}\\
{\it The critical value reflects the width of the top.}
\end{center}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf The width of Top of the Hebbian Peak.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=d-h-pat-dep.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf Conjecture 2.}\\
\hspace*{1in}\\
\hspace*{1in}\\
{\it
Traces of fitness values\\
after starting the downhill walk
\hspace*{1in}\\
\hspace*{1in}\\
reflects
\hspace*{1in}\\
\hspace*{1in}\\
the steepness of the sidewall.}
\end{center}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Sidewall of Two Different Hebbian Peaks.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=d-h-shape.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf A conclusion here:}\\
\hspace*{1in}\\
\hspace*{1in}\\
The hebbian peak has\\
\hspace*{1in}\\
a wider top and steeper sidewall \\
\hspace*{1in}\\
for smaller storage.
\end{center}

%%%%%%%%%%
\clearpage
\noindent
{\bf $\Box$ Question 2.}\\

{\it Does the peak shape affect the difficulty in search ?}\\
\begin{itemize}
\item[] We study this issue by observing:
   \begin{itemize}
   \item[$\star$] two different peaks found by EP:
      \begin{itemize}
      \item[$\cdot$] one is easy and the other is 
                            difficult to be found.
      \end{itemize}
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Two Evolutions by EP.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=ep.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Two Peaks Found by EP and the Hebbian Peak.}
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=fitness.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}
Our original paper submitted to ICNNAI'2001:\\
\hspace*{1in}\\
\hspace*{1in}\\
\hspace*{1in}\\
{\bf EP Downhill Walkers Explore a Landscape \\
\hspace*{1in}\\
in Weight Space\\ 
\hspace*{1in}\\
of a Fully-connected Neural Network Model:\\
\hspace*{1in}\\
\hspace*{1in}\\
Why does Hebbian Peak Resist to be Found?}
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf A conclusion here:}\\
\hspace*{1in}\\
\hspace*{1in}\\
EP can locate only sharp peaks,\\
\hspace*{1in}\\
and\\
\hspace*{1in}\\
the sharper its peak, the easier to be found. 
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
{\bf Observation\\
\hspace*{1in}\\
of\\
\hspace*{1in}\\
Weight Space of Associative Memory Model\\
\hspace*{1in}\\
by\\
\hspace*{1in}\\
Dimension Reduction Techniques.
}
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
$\Box$ {\bf Techniques of observing high-D space:}\\
\hspace*{1in}\\
\hspace*{1in}\\
Visualization by dimension reduction\\
\hspace*{1in}\\
using\\
\hspace*{1in}\\
$\star$ Principal Component Analysis (PCA)\\
\hspace*{1in}\\
$\star$ Kohonen's Topology-preserving Map\\
\hspace*{1in}\\
or\\
\hspace*{1in}\\

$\star$ Sammon Mapping
\end{center}

%%%%%%%%%%
\clearpage
\begin{center}
$\Box$ {\bf Principal Component Analysis (PCA)}\\
\hspace*{1in}\\
\hspace*{1in}\\
The goal is \\
\hspace*{1in}\\
to search for one axis (or a few multiple axes)\\
\hspace*{1in}\\
such that \\
\hspace*{1in}\\
we have the maximum variance of the projections \\
of the original points onto the axis.
\end{center}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Convergence behavior of an EC by PCA}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=pca.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{large}\noindent
$\Box$ {\bf Kohonen's Topology-preserving Map.}\\
\end{large}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=triangle.eps,width=15cm,height=10cm}
   \hspace*{1in}\\
   \hspace*{1in}\\
   \hspace*{1in}\\
   \epsfile{file=kohonen.eps,width=5.9cm,height=5.9cm}
 \end{center}
\end{figure}

%%%%%%%%%%
%%\clearpage
%%\begin{center}\begin{large}
%%{\bf Topology-preserving Map\\
%%from 2401-D weight Space to 2-D Coordinate:}\\
%%\hspace*{1in}\\
%%\end{large} \end{center}
%%\begin{figure}[htbp]
%% \begin{center}
%%   \epsfile{file=kohonen-2.eps,width=6cm,height=6cm}
%% \end{center}
%%\end{figure}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf Sammon Mapping} (J.\ W.\ Sammon, 1969)\\

\begin{itemize}
\item Assume $N$ points are given in the $n$-D space.\\
\item Calculate distance matrix $R$ ($N \times N$) whose $i$-$j$ element
          is the Euclidean distance between the $i$-th and $j$-th point.\\
\item Also think of a tentative $N$ points in the 2-D space that are
          located at random at the beginning.\\
\item The distance matrix $Q$ is calculated in the same way as $R$.\\
\item Then the error matrix $P=R-Q$ is defined.\\
\item Search for the locations of $N$ points in the 2-D space
          that minimizes the sum of element $P$.\\
\end{itemize}
\noindent
This is an optimization problem which we can solve by GA, that is,\\

\begin{itemize}
\item $N$ points in $n$-D space $\Rightarrow$ $N$ points in 2-D space 
   \begin{itemize}
   \item[]{\it with the distance relation being preserved 
               as much as possible;} or\\
   \end{itemize}
\item The $n$-D distances are approximated by 2-D distances 
   \begin{itemize}
   \item[] {\it with a minimal error.}
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
$\Box$ {\bf GA Implementation of Sammon Mapping}\\
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=summon.eps,width=18cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}\begin{large}
{\bf Normalization of Solutions on \\Hypersphere Surface:}\\
\end{large} \end{center}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=sphere.eps,width=6cm,height=6cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=line-point.eps,width=7.0cm,height=7.0cm}
   \epsfile{file=block-point.eps,width=7.0cm,height=7.0cm}
 \end{center}
\end{figure}
\hspace*{1in}\\
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=sphere-two.eps,width=7.0cm,height=7.0cm}
   \epsfile{file=sphere-one-small.eps,width=7.0cm,height=7.0cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=hopfield-point-10-new.eps,width=12cm,height=12cm}
   \epsfile{file=hopfield-point-90.eps,width=12cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%%
\clearpage
\begin{center}
{\large 100 solutions + 100 points near the origin.}
\end{center}
\begin{figure}[htbp]
 \begin{center}
   \epsfile{file=many-origins.eps,width=12cm,height=12cm}
 \end{center}
\end{figure}

%%%%%%%%%% Conclusion %%%%%%%%%%
\clearpage
{\bf $\Box$ Conclusions:}\\
\begin{itemize}
\item We have many different weight solutions for Associative Memory,
   \begin{itemize}
   \item where questions are:
      \begin{itemize}
      \item How do they look in the fitness landscape? 
      \item Can we learn the distribution of them
            in weight space?
         \begin{itemize}
         \item which is still one of the open issues.
         \end{itemize}
      \end{itemize}
   \end{itemize}
\item[]
\item The answer is (somehow) yes!
   \begin{itemize}
   \item That is:
      \begin{itemize}
      \item EP downhill walkers 
         \begin{itemize}
         \item[$\cdot$] reveal the shape of the hills;
         \end{itemize}
      \item Dimengin reduction techniqus visualize 
         \begin{itemize}
         \item[$\cdot$] a high dimensional space.
         \end{itemize}
      \end{itemize}
   \end{itemize}
\item[]
\item We shed new light on the classical unknown issues.
\item Can we apply these techniques to real world problems too?
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
{\bf $\Box$ Related Web Pages:}\\

\begin{itemize}
\item {\bf Our papers on the topics of this talk:}
   \begin{itemize}
   \item http://anka.cs.deu.edu.tr/~akira/myresearch.html
   \end{itemize}
\item[]
\item {\bf Slides in this talk:}
   \begin{itemize}
   \item http://anka.cs.deu.edu.tr/~akira/icnnai-2001-talk.html
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\clearpage
\noindent
{\bf $\Box$ Web Pages of Interest:}\\

\begin{itemize}
\item {\bf GA-Digest}
   \begin{itemize}
   \item http://www.aic.nrl.navy.mil/galist
      \begin{itemize}
      \item To subscribe send email to 
          \begin{itemize}
          \item listproc@gmu.edu 
          \end{itemize}
      \item containing the following text in the body:
          \begin{itemize}
          \item subscribe ga-list $<$Your Name$>$ \\
          \end{itemize}
      \end{itemize}
   \end{itemize}
\item {\bf Illinois Genetic Algorithms Laboratory (IlliGAL)}
   \begin{itemize}
   \item http://www-illigal.ge.uiuc.edu \\
   \end{itemize}
\item {\bf Genetic Programming Notebook}
   \begin{itemize}
   \item http://www.geneticprogramming.com \\
   \end{itemize}
\item {\bf Alife Online}
   \begin{itemize}
   \item http://alife.org
   \end{itemize}
\end{itemize}

%%%%%%%%%%
\end{LARGE}
\end{document}
