Lower bounds on the restitution error in the Hopfield model
This paper analyzes the question of the storage capacity of the Hopfield model. Non-rigorous analytic as well as numerical results suggest that the Hopfield model with \(N\) neurons can store up \(\alpha_cN\) patterns, where \(\alpha_c\approx 0.14\). While it has been proven by \textit{Ch. M. Newman} [Networks 1, 223-238 (1988)] that \(\alpha_c\geq 0.055\), there is no rigorous result that would show that \(\alpha_c<\infty\). The present paper takes a step into this direction by studying the local minima of the Hamiltonian in the neighborhood of one pattern. It is proven that for any \(\alpha>0\), there exists a \(\delta(\alpha)>0\), such that in a ball of radius \(\delta(\alpha)\) (distances are measured by the Hamming distance), there is no local minimum. Furthermore, it is shown than as \(\alpha\) tends to infinity, \(\delta(\alpha)\) becomes larger that 0.05. In spite of the relatively poor numerical result, the paper is interesting to read and contains a number of very nice and original ideas.
scientific article; zbMATH DE number 1002142