From 18337093ce5063470097da0f6759e21b36d00a2b Mon Sep 17 00:00:00 2001 From: =?utf8?q?Jean-Fran=C3=A7ois=20Couchot?= Date: Tue, 16 Jun 2015 15:28:24 +0200 Subject: [PATCH] =?utf8?q?d=C3=A9but=20ANN=20chaotique=20=3F?= MIME-Version: 1.0 Content-Type: text/plain; charset=utf8 Content-Transfer-Encoding: 8bit --- chaosANN.tex | 66 +++++++++++++++++++++++++-------------------------- main.pdf | Bin 1979020 -> 1975663 bytes 2 files changed, 33 insertions(+), 33 deletions(-) diff --git a/chaosANN.tex b/chaosANN.tex index cdcd912..cb1ce2a 100644 --- a/chaosANN.tex +++ b/chaosANN.tex @@ -164,43 +164,43 @@ des itération unaires chaotiques?} Cette section s'intéresse à étudier le comportement d'un réseau de neurones face à des itérations unaires chaotiques, comme définies à la section~\ref{sec:TIPE12}. +Plus précésment, on considère dans cette partie une fonction dont le graphe +des itérations unaires est fortement connexe et une séquence dans +$[n]^{\mathds{N}}$. On cherche à construire un réseau de neurones +qui approximerait les itérations de la fonction $G_{f_u}$ comme définie +à l'équation~(\ref{eq:sch:unaire}). +Sans perte de généralité, on considère dans ce qui suit une instance +de de fonction à quatre éléments. -\subsection{Representing Chaotic Iterations for Neural Networks} +\subsection{Construction du réseau} \label{section:translation} -The problem of deciding whether classical feedforward ANNs are -suitable to approximate topological chaotic iterations may then be -reduced to evaluate such neural networks on iterations of functions -with Strongly Connected Component (SCC)~graph of iterations. To -compare with non-chaotic iterations, the experiments detailed in the -following sections are carried out using both kinds of function -(chaotic and non-chaotic). Let us emphasize on the difference between -this kind of neural networks and the Chaotic Iterations based -multilayer peceptron. - -We are then left to compute two disjoint function sets that contain -either functions with topological chaos properties or not, depending -on the strong connectivity of their iterations graph. This can be -achieved for instance by removing a set of edges from the iteration -graph $\Gamma(f_0)$ of the vectorial negation function~$f_0$. One can -deduce whether a function verifies the topological chaos property or -not by checking the strong connectivity of the resulting graph of -iterations. - -For instance let us consider the functions $f$ and $g$ from $\Bool^4$ -to $\Bool^4$ respectively defined by the following lists: -$$[0, 0, 2, 3, 13, 13, 6, 3, 8, 9, 10, 11, 8, 13, 14, - 15]$$ $$\mbox{and } [11, 14, 13, 14, 11, 10, 1, 8, 7, 6, 5, 4, 3, 2, - 1, 0] \enspace.$$ In other words, the image of $0011$ by $g$ is -$1110$: it is obtained as the binary value of the fourth element in -the second list (namely~14). It is not hard to verify that -$\Gamma(f)$ is not SCC (\textit{e.g.}, $f(1111)$ is $1111$) whereas -$\Gamma(g)$ is. The remaining of this section shows how to translate -iterations of such functions into a model amenable to be learned by an -ANN. Formally, input and output vectors are pairs~$((S^t)^{t \in - \Nats},x)$ and $\left(\sigma((S^t)^{t \in - \Nats}),F_{f}(S^0,x)\right)$ as defined in~Eq.~(\ref{eq:Gf}). +On considère par exemple les deux fonctions $f$ and $g$ de0 $\Bool^4$ +dans $\Bool^4$ définies par: + +\begin{eqnarray*} +f(x_1,x_2,x_3,x_4) &= & +(x_1(x_2+x_4)+ \overline{x_2}x_3\overline{x_4}, +x_2, +x_3(\overline{x_1}.\overline{x_4}+x_2x_4+x_1\overline{x_2}), +x_4+\overline{x_2}x_3) \\ +g(x_1,x_2,x_3,x_4) &= & +(\overline{x_1}, +\overline{x_2}+ x_1.\overline{x_3}.\overline{x_4}, +\overline{x_3}(x_1 + x_2+x_4), +\overline{x_4}(x_1 + \overline{x_2}+\overline{x_3})) +\end{eqnarray*} +On peut vérifier facilement que le graphe $\textsc{giu}(f)$ +n'est pas fortement connexe car $(1,1,1,1)$ est un point fixe de $f$ +tandis que le graphe $\textsc{giu}(g)$ l'est. + +L'entrée du réseau est une paire de la forme +$(x,(S^t)^{t \in \Nats})$ et sa sortie correspondante est +de la forme $\left(F_{h_u}(S^0,x), \sigma((S^t)^{t \in + \Nats})\right)$ comme définie à l'équationà l'équation~(\ref{eq:sch:unaire}). + + Firstly, let us focus on how to memorize configurations. Two distinct translations are proposed. 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