Convergence of a learning rule with generalized hebbian synapses
by
Fernanda Botelho
University of Memphis
Coauthors: J. Jamison

This work studies the convergence properties of a learning model with generalized hebbian synapses. This learning model relies upon synaptic adjustments that incorporate a probabilistic replication at the synaptic level and a nonlinear correction term. These two features are captured by a sub-stochastic replication tridiagonal matrix T and a third degree nonlinear term, respectively. The learning algorithm halts if and only if the initial condition is a solution of a matrix equation: TCw -(w.Cw)w=0, where T is a tridiagonal n-matrix of probabilities, C is a symmetric n-matrix of expected values (each entry is known as a connecting weight of a network architecture implementing the learning model), w.Cw represents the standard inner product in the nth euclidean space. Our main goal is to solve this matrix equation. This was done in two steps, first assuming T positive definite and next T not positive definite. In this second case, the solution set was found under an additional symmetry condition. The last part of this paper is concerned with the stability study of each solution. We found that, under some constrains, the stable solutions are the principal directions of a natural product of square root of T and C. These observations allow us to conclude that such a model may act as an information filter.

Date received: January 11, 2002