From 3e9929bb13d395a475ddfd4045cb00b3be5caa7f Mon Sep 17 00:00:00 2001 From: =?utf8?q?Jean-Fran=C3=A7ois=20Couchot?= Date: Tue, 13 Sep 2011 16:10:08 +0200 Subject: [PATCH] review 1 --- main.tex | 42 +++++++++++++++++++++++++++++++----------- scheme.odg | Bin 0 -> 10791 bytes scheme.pdf | Bin 0 -> 27531 bytes 3 files changed, 31 insertions(+), 11 deletions(-) create mode 100644 scheme.odg create mode 100644 scheme.pdf diff --git a/main.tex b/main.tex index 717ae3e..c134123 100644 --- 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} -\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 @@ -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} -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 $S \in \llbracket 1;n -\rrbracket^{\mathds{N}}$, 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 $\left(S,(x_1^0,\dots, x_n^0)\right) \in \llbracket 1;n -\rrbracket^{\mathds{N}} \times \mathds{B}^n$. 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''. @@ -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} -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. + + + + + + \section{Suitability of Artificial Neural Networks for Predicting Chaotic Behaviors} diff --git a/scheme.odg b/scheme.odg new file mode 100644 index 0000000000000000000000000000000000000000..8debe12534f297b13f55ca4d8a5e8f31d05d36cf GIT binary patch literal 10791 zcmb7q1wd3y_ctINA_z!#BMs7>N+S(Yy9+F^?7F)&A|joFfRuoM(k-o&ASERw-KnI~ zh~J_Qk3P?<@Av)(cK6=7=ltf(oVjy<=gigBxP(cDhK7rV_QHi%F8vZ;D-RkP+W7}n zg$4mbKuAv*$OZ;;1_Nx6V5kefn+uQ+YJ&hH_@FS53lIu$g@9a;eD-h~m;)Gq(ESdG zi;Mdm5QXt~AwV4gpf1RBX%C3=hS`M6l&HA${iIDRp^Q|6;8sB?I| zoocQ^D0#qtF><)DMCLKVkkoSXedn@%E%Z(-L}J)v^|JHM!k47y@EJ)v4`I|fM((}O z5>FjpE#=+F@$)j86&ee-O<_OEadI_HXnQAcBPFt-%(aUG8~;uJqbx|1UV(9;33ugu z2O}5BD^V|5|RB8?i=C7pfFJ~VIv9a06?5$>d1q|k@q}sL1Uy=d(1*wi?E>5Vh`pvRn zC+HDXcS+N`em$AmPakg{Kp>5NpJR5k3KL${ZXu0!^F&IB!L{pA(fNwvRp=Xp$)Q!+9m|oDAD(Jov#T*q zEFVr6j=8mwsi8hzxNTG&G1s6@Qw?C;wvrCsW>OArt70@3(2^Kl0~nYd@(eF zbDpE`m|jo&xScUAk`tQm4a*{*HM|-4S^e<@dSh*V71mLn!F>YU;^7*3N8UsL=Qp~L 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