Neural Associative Memory and the Willshaw–Palm Probability Distribution

Abstract

Previous asymptotic analyses of binary neural associative networks of Willshaw or Steinbuch type relied on a binomial approximation of the neurons' dendritic potentials. This approximation has been proven to be good only if the stored patterns are extremely sparse, for example, when the mean number of active units k per pattern vector scales with the logarithm of the vector size n. Recent promising results concerning storage capacity and retrieval efficiency for larger pattern activities $k > \log n$ have been doubted because here the binomial approximation can lead to a massive overestimation of performance. In this work I compute and characterize the exact Willshaw–Palm distribution of the dendritic potentials for hetero-association, auto-association, and fixed and random pattern activity. Comparing the raw and central moments of the Willshaw–Palm distribution to the moments of the corresponding binomial probability reveals that, asymptotically, the binomial approximation becomes exact for almost any sublinear pattern activity, including $k = O(n/\log^2 n)$. This verifies, for large networks, the existence of a wide high-performance parameter range as predicted by the approximative theory.