fields of application. Unfortunately, up to now these networks are
usually claimed to be chaotic without any mathematical proof. The
purpose of this paper is to establish, based on a rigorous theoretical
-framework, an equivalence between chaotic iterations according to the
-Devaney's formulation of chaos and a particular class of neural
+framework, an equivalence between chaotic iterations according to
+Devaney and a particular class of neural
networks. On the one hand we show how to build such a network, on the
other hand we provide a method to check if a neural network is a
chaotic one. Finally, the ability of classical feedforward multilayer
perceptrons to learn sets of data obtained from a chaotic dynamical
system is regarded. Various Boolean functions are iterated on finite
-states, some of them are proven to be chaotic as it is defined by
+states. Iterations of some of them are proven to be chaotic
+ as it is defined by
Devaney. In that context, important differences occur in the training
process, establishing with various neural networks that chaotic
behaviors are far more difficult to learn.
\maketitle
\begin{quotation}
+
Chaotic neural networks have received a lot of attention due to the
appealing properties of deterministic chaos (unpredictability,
sensitivity, and so on). However, such networks are often claimed
What all of these chaotic neural networks have in common is that they
are claimed to be chaotic despite a lack of any rigorous mathematical
-proof. The first contribution of this paper is to fill this gap, using a
-theoretical framework based on the Devaney's definition of chaos
+proof. The first contribution of this paper is to fill this gap,
+using a theoretical framework based on the Devaney's definition of chaos
\cite{Devaney}. This mathematical theory of chaos provides both
qualitative and quantitative tools to evaluate the complex behavior of
a dynamical system: ergodicity, expansivity, and so on. More
discrete chaotic dynamical systems. This family, called chaotic
iterations, is defined by a Boolean vector, an update function, and a
sequence giving which component to update at each iteration. It has
-been previously established that such dynamical systems can behave
-chaotically, as it is defined by Devaney, when the chosen function has
+been previously established that such dynamical systems is
+chaotically iterated (as it is defined by Devaney) when the chosen function has
a strongly connected iterations graph. In this document, we
experiment several MLPs and try to learn some iterations of this kind.
We show that non-chaotic iterations can be learned, whereas it is
due to their inability to learn chaotic behaviors in this context.
The remainder of this research work is organized as follows. The next
-section is devoted to the basics of chaotic iterations and Devaney's
+section is devoted to the basics of Devaney's
chaos. Section~\ref{S2} formally describes how to build a neural
network that operates chaotically. Section~\ref{S3} is
devoted to the dual case of checking whether an existing neural network
sets. The paper ends with a conclusion section where our contribution
is summed up and intended future work is exposed.
-\section{Link between Chaotic Iterations and Devaney's Chaos}
+\section{Chaotic Iterations according to Devaney}
In this section, the well-established notion of Devaney's mathematical
chaos is firstly recalled. Preservation of the unpredictability of
such dynamical system when implemented on a computer is obtained by
-using some discrete iterations called ``chaotic iterations'', which
-are thus introduced. The result establishing the link between chaotic
+using some discrete iterations called ``asynchronous iterations'', which
+are thus introduced. The result establishing the link between such
iterations and Devaney's chaos is finally presented at the end of this
section.
Finally,
\begin{definition} \label{def5}
-$f$ is {\bf chaotic according to Devaney} on $(\mathcal{X},\tau)$ if
- $f$ is regular, topologically transitive, and has sensitive
- dependence to its initial conditions.
+The dynamical system that iterates $f$ is {\bf chaotic according to Devaney}
+on $(\mathcal{X},\tau)$ if $f$ is regular, topologically transitive,
+and has sensitive dependence to its initial conditions.
\end{definition}
Let us notice that for a metric space the last condition follows from
the two first ones~\cite{Banks92}.
-\subsection{Chaotic Iterations}
+\subsection{Asynchronous Iterations}
-This section presents some basics on topological chaotic iterations.
+%This section presents some basics on topological chaotic iterations.
Let us firstly discuss about the domain of iteration. As far as we
-know, no result rules that the chaotic behavior of a function that has
-been theoretically proven on $\R$ remains valid on the floating-point
+know, no result rules that the chaotic behavior of a dynamical system
+that has been theoretically proven on $\R$ remains valid on the
+floating-point
numbers, which is the implementation domain. Thus, to avoid loss of
chaos this work presents an alternative, that is to iterate Boolean
maps: results that are theoretically obtained in that domain are
preserved in implementations.
Let us denote by $\llbracket a ; b \rrbracket$ the following interval
-of integers: $\{a, a+1, \hdots, b\}$, where $a~<~b$. A {\it system}
+of integers: $\{a, a+1, \hdots, b\}$, where $a~<~b$.
+In that section, a system
under consideration iteratively modifies a collection of
$n$~components. Each component $i \in \llbracket 1; n \rrbracket$
takes its value $x_i$ among the domain $\Bool=\{0,1\}$. A~{\it
- configuration} of the system at discrete time $t$ (also said at {\it
- iteration} $t$) is the vector
+ configuration} of the system at discrete time $t$ is the vector
%\begin{equation*}
$x^{t}=(x_1^{t},\ldots,x_{n}^{t}) \in \Bool^n$.
%\end{equation*}
$i$ from $x$ to $N(i,x)$ if and only if $f_i(x)$ is $N(i,x)$.
In the sequel, the {\it strategy} $S=(S^{t})^{t \in \Nats}$ is the
-sequence defining the component to update at time $t$ and $S^{t}$
-denotes its $t-$th term. We introduce the function $F_{f}$ that is
+sequence defining which component to update at time $t$ and $S^{t}$
+denotes its $t-$th term.
+This iteration scheme that only modifies one element at each iteration
+is clasically refered as \emph{asynchronous iterations}.
+Next section shows the link between asynchronous iterations and
+Devaney's Chaos.
+
+\subsection{On the link between asynchronous iterations and
+ Devaney's Chaos}
+
+In this subsection we recall the link we have established between
+asynchronous iterations and Devaney's chaos. The theoretical framework is
+fully described in \cite{guyeux09}.
+
+We introduce the function $F_{f}$ that is
defined for any given application $f:\Bool^{n} \to \Bool^{n}$ by
$F_{f}: \llbracket1;n\rrbracket\times \mathds{B}^{n} \rightarrow
\mathds{B}^{n}$, s.t.
where $\sigma$ is the function that removes the first term of the
strategy ({\it i.e.},~$S^0$). This definition means that only one
component of the system is updated at an iteration: the $S^t$-th
-element. But it can be extended by considering subsets for $S^t$.
+element. But it can be extended by considering subsets for $S^t$.
-Let us finally remark that, despite the use of the term {\it chaotic},
-there is {\it priori} no connection between these iterations and the
-mathematical theory of chaos presented previously.
-\subsection{Chaotic Iterations and Devaney's Chaos}
-In this subsection we recall the link we have established between
-chaotic iterations and Devaney's chaos. The theoretical framework is
-fully described in \cite{guyeux09}.
The {\bf distance} $d$ between two points $(S,x)$ and
$(\check{S},\check{x})\in \mathcal{X} = \llbracket1;n\rrbracket^\Nats
link, Guyeux~\cite{GuyeuxThese10} has proven that,
\begin{theorem}%[Characterization of $\mathcal{C}$]
\label{Th:Caracterisation des IC chaotiques}
-Let $f:\Bool^n\to\Bool^n$. $G_f$ is chaotic (according to Devaney)
-if and only if $\Gamma(f)$ is strongly connected.
+Let $f:\Bool^n\to\Bool^n$. Iterations of $G_f$ are chaotic according
+to Devaney if and only if $\Gamma(f)$ is strongly connected.
\end{theorem}
Checking if a graph is strongly connected is not difficult. For
chaotic iterations of $F_f$ with initial
condition $\left(S,(x_1^0,\dots, x_n^0)\right) \in \llbracket 1;n
\rrbracket^{\mathds{N}} \times \mathds{B}^n$.
-Theoretically speakig, such iterations of $F_f$ are thus a formal model of
+Theoretically speaking, 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''.
\begin{definition} \label{def10}
A dynamical system $\left( \mathcal{X}, f\right)$ is {\bf
-indecomposable} if it is not the union of two closed sets $A, B
+not decomposable} if it is not the union of two closed sets $A, B
\subset \mathcal{X}$ such that $f(A) \subset A, f(B) \subset B$.
\end{definition}
\begin{figure}
\centering
\includegraphics[scale=0.625]{scheme}
- \caption{Summary of addressed membership problems}
+ \caption{Summary of addressed neural networks and chaos problems}
\label{Fig:scheme}
\end{figure}
-The Figure~\ref{Fig:scheme} is a summary of the addressed problems.
+The Figure~\ref{Fig:scheme} is a summary of addressed neural networks and chaos 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
property. Furthermore, MLPs have been already considered for chaotic
time series prediction. For example, in~\cite{dalkiran10} the authors
have shown that a feedforward MLP with two hidden layers, and trained
-with Bayesian Regulation backpropagation, can learn successfully the
+with Bayesian Regulation back-propagation, can learn successfully the
dynamics of Chua's circuit.
-In these experimentations we consider MLPs having one hidden layer of
+In these experiments we consider MLPs having one hidden layer of
sigmoidal neurons and output neurons with a linear activation
function. They are trained using the Limited-memory
Broyden-Fletcher-Goldfarb-Shanno quasi-newton algorithm in combination
