Abstract
We consider the Hopfield model with N neurons and an increasing number M = M(N) of randomly chosen patterns. Under the condition M 2/N → 0, we prove for every fixed choice of overlap parameters a central limit theorem as N → ∞, 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 M 3/2 log M ≤ N suffices. As in the case of a finite number of patterns, the central limit theorem requires a centering which depends on the random patterns. In addition, we describe the almost sure asymptotic behavior of the partition function under the condition M 3/N → 0.
Research supported by the Swiss National Foundation, Contract No. 21-298333.90
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References
Bovier, A., V. Gayrard, and P. Picco, “Gibbs states of the Hopfield model in the regime of perfect memory”, Probab. Theory Relat. Fields 100, (1994), 329–363.
Bovier, A., and V. Gayrard, “An almost sure large deviation principle for the Hopfield model,” Ann. Probab. 24, (1996), 1444–1475.
Bovier, A., and V. Gayrard, “The retrieval phase of the Hopfield model: a rigorous analysis of the overlap distribution”, Probab. Theory Relat. Fields 107, (1997), 61–98.
Bovier, A., and V. Gayrard, “Hopfield models as generalized random mean field models,” in Mathematics of Spin Glasses and Neural Networks (Bovier, A., and P. Picco, eds.), Birkhäuser, Boston, 1998.
Bovier, A., and V. Gayrard, “An almost sure central limit theorem for the Hopfield model,” preprint (1996).
Ellis, R.S., Entropy, Large Deviations, and Statistical Mechanics, Grundlehren der mathematischen Wissenschaften, vol. 271, Springer-Verlag, New York, 1985.
Gentz, B., “An almost sure central limit theorem for the overlap parameters in the Hopfield model,” Stochastic Process. Appl. 62, (1996), 243–262.
Gentz, B., “A central limit theorem for the overlap in the Hopfield model,” Ann. Probab. 24, (1996), 1809–1841.
Gentz, B., “A central limit theorem for the overlap in the Hopfield model,” Ph.D. Thesis, University of Zürich, 1996.
Ledoux, M., and M. Talagrand, Probability in Banach Spaces: Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1991.
Talagrand, M., “Concentration of Measure and Isoperimetric Inequalities in Product Spaces,” Publ. Math. IHES 81, (1995), 73–205.
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© 1998 Birkhäuser Boston
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Gentz, B. (1998). On the Central Limit Theorem for the Overlap in the Hopfield Model. In: Bovier, A., Picco, P. (eds) Mathematical Aspects of Spin Glasses and Neural Networks. Progress in Probability, vol 41. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4102-7_3
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DOI: https://doi.org/10.1007/978-1-4612-4102-7_3
Publisher Name: Birkhäuser Boston
Print ISBN: 978-1-4612-8653-0
Online ISBN: 978-1-4612-4102-7
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