Group actions and learning for a family of automata

摘要

We study properties of a family of neural network type of automata which are generalizations of the Hopfield model for content addressable memory using a novel approach based on the action of groups of isometries of the hypercube. We review storage algorithms and show that the classical sum of outerproducts scheme and several of its variations are invariant under such action. As a consequence, we prove that the space of states is partitioned into orbits of points having the same dynamical behavior. We find minimal sets of conditions forcing this property and discuss applications for the structure of the stable points. Calculations for a simple orthogonal case are given as an example.