Learning Systems Pattern Recognition Graph Theory Image Processing Recurrent Neural Networks

Issue Date:

2005

Publisher:

Institute of Information Theories and Applications FOI ITHEA

Abstract:

As is well known, the Convergence Theorem for the Recurrent Neural Networks, is based in
Lyapunov ́s second method, which states that associated to any one given net state, there always exist a real
number, in other words an element of the one dimensional Euclidean Space R, in such a way that when the state
of the net changes then its associated real number decreases. In this paper we will introduce the two dimensional
Euclidean space R2, as the space associated to the net, and we will define a pair of real numbers ( x, y ) ,
associated to any one given state of the net. We will prove that when the net change its state, then the product
x ⋅ y will decrease. All the states whose projection over the energy field are placed on the same hyperbolic
surface, will be considered as points with the same energy level. On the other hand we will prove that if the states
are classified attended to their distances to the zero vector, only one pattern in each one of the different classes
may be at the same energy level. The retrieving procedure is analyzed trough the projection of the states on that
plane. The geometrical properties of the synaptic matrix W may be used for classifying the n-dimensional state-
vector space in n classes. A pattern to be recognized is seen as a point belonging to one of these classes, and
depending on the class the pattern to be retrieved belongs, different weight parameters are used. The capacity of
the net is improved and the spurious states are reduced. In order to clarify and corroborate the theoretical results,
together with the formal theory, an application is presented.