Retrieval Properties of Neural Networks with Infinitely Many Marked Patterns

Abstract

We study an extension of the Hopfield model, where a certain number of “privileged” (or “marked”) patterns enters the Hebb rule with weight 1, while the remaining enter with weight γ < 1. Both sets are infinite. Assuming replica symmetry, the zero-temperature capacity vs. γ and overlap vs. γ diagrams are calculated and compared with numerical simulations, when both sets hold the same number of patterns. The transitions from the retrieval to the spin glass phase are found to be first order.