Part of Advances in Neural Information Processing Systems 26 (NIPS 2013)
We solve the mean field equations for a stochastic Hopfield network with temperature (noise) in the presence of strong, i.e., multiply stored patterns, and use this solution to obtain the storage capacity of such a network. Our result provides for the first time a rigorous solution of the mean field equations for the standard Hopfield model and is in contrast to the mathematically unjustifiable replica technique that has been hitherto used for this derivation. We show that the critical temperature for stability of a strong pattern is equal to its degree or multiplicity, when sum of the cubes of degrees of all stored patterns is negligible compared to the network size. In the case of a single strong pattern in the presence of simple patterns, when the ratio of the number of all stored patterns and the network size is a positive constant, we obtain the distribution of the overlaps of the patterns with the mean field and deduce that the storage capacity for retrieving a strong pattern exceeds that for retrieving a simple pattern by a multiplicative factor equal to the square of the degree of the strong pattern. This square law property provides justification for using strong patterns to model attachment types and behavioural prototypes in psychology and psychotherapy.