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A central limit theorem for the overlap in the Hopfield model


Gentz, B (1996). A central limit theorem for the overlap in the Hopfield model. The Annals of Probability, 24(4):1809-1841.

Abstract

We consider the Hopfield model with n neurons and an increasing number $p = p(n)$ of randomly chosen patterns. Under the condition $(p^3 \log p)/n \to 0$, we prove for every fixed choice of overlap parameters a central limit theorem as $n \to \infty$, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition $(p^2 \log p)/n \to 0$ suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.

Abstract

We consider the Hopfield model with n neurons and an increasing number $p = p(n)$ of randomly chosen patterns. Under the condition $(p^3 \log p)/n \to 0$, we prove for every fixed choice of overlap parameters a central limit theorem as $n \to \infty$, which holds for almost all realizations of the random patterns. In the special case where the temperature is above the critical one and there is no external magnetic field, the condition $(p^2 \log p)/n \to 0$ suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns.

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7 citations in Web of Science®
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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Fluctuations; Hopfield model; overlap; neural networks; Laplace's method
Language:English
Date:1996
Deposited On:29 Nov 2010 16:28
Last Modified:05 Apr 2016 13:27
Publisher:Institute of Mathematical Statistics
ISSN:0091-1798
Publisher DOI:https://doi.org/10.1214/aop/1041903207
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1415230
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0872.60015

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