review 1

[chaos1.git] / main.tex

diff --git a/main.tex b/main.tex
index 717ae3e688d7b36d00dcd58280383073825e8c46..c134123771b8a29154e07475414f153722d706fe 100644 (file)
--- a/main.tex
+++ b/main.tex
@@ -503,7 +503,7 @@ bits.   Moreover, each  binary  output is  connected  with a  feedback
 connection to an input one.
 
 \begin{itemize}
 connection to an input one.
 
 \begin{itemize}

-\item During  initialization, the network if we seed it with $n$~bits denoted
+\item During  initialization, the network is seeded with $n$~bits denoted

   $\left(x^0_1,\dots,x^0_n\right)$  and an  integer  value $S^0$  that
   belongs to $\llbracket1;n\rrbracket$.
 \item     At     iteration~$t$,     the     last     output     vector
   $\left(x^0_1,\dots,x^0_n\right)$  and an  integer  value $S^0$  that
   belongs to $\llbracket1;n\rrbracket$.
 \item     At     iteration~$t$,     the     last     output     vector


@@ -529,12 +529,15 @@ $f\left(x_1,x_2,\dots,x_n\right)$ is equal to
 \left(F\left(1,\left(x_1,x_2,\dots,x_n\right)\right),\dots,
   F\left(n,\left(x_1,x_2,\dots,x_n\right)\right)\right) \enspace .
 \end{equation}
 \left(F\left(1,\left(x_1,x_2,\dots,x_n\right)\right),\dots,
   F\left(n,\left(x_1,x_2,\dots,x_n\right)\right)\right) \enspace .
 \end{equation}

-Then $F=F_f$  and this recurrent  neural network produces  exactly the
-same      output      vectors,      when     feeding      it      with
+Then $F=F_f$. If this recurrent  neural network is seeded with 

 $\left(x_1^0,\dots,x_n^0\right)$    and   $S   \in    \llbracket   1;n
 $\left(x_1^0,\dots,x_n^0\right)$    and   $S   \in    \llbracket   1;n

-\rrbracket^{\mathds{N}}$, than  chaotic iterations $F_f$  with initial
+\rrbracket^{\mathds{N}}$, it produces  exactly the
+same      output      vectors  than the 
+chaotic iterations of $F_f$  with initial

 condition  $\left(S,(x_1^0,\dots,  x_n^0)\right)  \in  \llbracket  1;n
 condition  $\left(S,(x_1^0,\dots,  x_n^0)\right)  \in  \llbracket  1;n

-\rrbracket^{\mathds{N}}  \times \mathds{B}^n$.   In the  rest  of this
+\rrbracket^{\mathds{N}}  \times \mathds{B}^n$.
+Theoretically speakig, such iterations of $F_f$ are thus a formal model of  
+these kind of recurrent neural networks. In the  rest  of this

 paper,  we will  call such  multilayer perceptrons  CI-MLP($f$), which
 stands for ``Chaotic Iterations based MultiLayer Perceptron''.
 
 paper,  we will  call such  multilayer perceptrons  CI-MLP($f$), which
 stands for ``Chaotic Iterations based MultiLayer Perceptron''.
 


@@ -652,15 +655,32 @@ $\left( \mathcal{X},d\right)$  is compact and  the topological entropy
 of $(\mathcal{X},G_{f_0})$ is infinite.
 \end{theorem}
 
 of $(\mathcal{X},G_{f_0})$ is infinite.
 \end{theorem}
 

-We have explained how to  construct truly chaotic neural networks, how
-to check whether a  given MLP is chaotic or not, and  how to study its
-topological behavior.   The last thing to  investigate, when comparing
-neural  networks   and  Devaney's  chaos,  is   to  determine  whether
-artificial neural networks  are able to learn or  predict some chaotic
-behaviors, as  it is defined  in the Devaney's formulation  (when they
+\begin{figure}
+  \centering
+  \includegraphics[scale=0.625]{scheme}
+  \caption{Summary of addressed membership problems}
+  \label{Fig:scheme}
+\end{figure}
+
+The Figure~\ref{Fig:scheme} is a summary of the addressed problems.
+Section~\ref{S2} has explained how to  construct a truly chaotic neural
+networks $A$ for instance.
+Section~\ref{S3} has shown how to check whether a  given MLP
+$A$ or $C$ is chaotic or not in the sens of Devaney.
+%, and  how to study its topological behavior. 
+The last thing to  investigate, when comparing
+neural  networks   and  Devaney's  chaos,  is to  determine  whether
+an artificial neural network $A$  is able to learn or  predict some chaotic
+behaviors of $B$, as  it is defined  in the Devaney's formulation  (when they

 are not specifically constructed for this purpose).  This statement is
 studied in the next section.
 
 are not specifically constructed for this purpose).  This statement is
 studied in the next section.
 

+
+
+
+
+
+

 \section{Suitability of Artificial Neural Networks 
 for Predicting Chaotic Behaviors}
 
 \section{Suitability of Artificial Neural Networks 
 for Predicting Chaotic Behaviors}