Associative Memories in Infinite Dimensional Spaces

A generalization of the Little–Hopfield neural network model for associative memories is presented that considers the case of a continuum of processing units. The state space corresponds to an infinite dimensional euclidean space. A dynamics is proposed that minimizes an energy functional that is a natural extension of the discrete case. The case in which the synaptic weight operator is defined through the autocorrelation rule (Hebb rule) with orthogonal memories is analyzed. We also consider the case of memories that are not orthogonal. Finally, we discuss the generalization of the non deterministic, finite temperature dynamics.