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Sharp upper bounds on perfect retrieval in the Hopfield model

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes

Published online by Cambridge University Press:  14 July 2016

Anton Bovier
Anton Bovier*
Affiliation:
Weierstraß–Institut für Angewandte Analysis und Stochastik
*
∗Postal address: WIAS, Mohrenstrasse 39, D-10117 Berlin, Germany. Email address: bovier@wias-berlin.de
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Abstract

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 Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

Keywords

Hopfield modelstorage capacitygradient dynamicssequential dynamics

MSC classification

Primary: 82C32: Neural nets 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 941 - 950
Copyright
Copyright © Applied Probability Trust 1999 

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