Competition between Hopfield and symmetry transform interactions in a neural net
M. R. Evans, D. J. Wallace, C Zhan · Journal of Physics A Mathematical and General · 1991
The authors consider a neural network proposed by Coolen and Kuijk (1989) that has interactions composed of two competing elements. The first is the Hopfield interaction for recall of a set of stored patterns. The second is interactions between pairs of sites arranged so that the configuration undergoes a symmetry transformation at each parallel update. First they consider the effect of the symmetry-transform interactions alone. They show that for sequential updating the symmetry transformation is no longer carried out faithfully, but rather the spin configuration tends to a symmetry invariant. In order to understand how the retrieval phase of the Hopfield model is disrupted by the symmetry-transform interactions they perform a replica symmetric analysis. They demonstrate that the symmetry-transform interactions generate a noise very similar to that of random external fields on the memory states. The phase diagram suggests the possibility of symmetry-invariant recognition for an extensive number of patterns and an optimal value for the symmetry-transform interaction strength. They present numerical simulations of the model under parallel dynamics to confirm these predictions.