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Sharp upper bounds on perfect retrieval in the Hopfield model. (English) Zbl 0947.60092

Summary: We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of D. Loukianova [Probab. Theory Relat. Fields 107, No. 2, 161-176 (1997; Zbl 0870.60021)] using the negative association properties of some random variables arising in the analysis.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
91E40 Memory and learning in psychology
82C32 Neural nets applied to problems in time-dependent statistical mechanics

Citations:

Zbl 0870.60021
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