The Annals of Probability

A central limit theorem for the overlap in the Hopfield model

Barbara Gentz

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 1809-1841.

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903207

Digital Object Identifier
doi:10.1214/aop/1041903207

Mathematical Reviews number (MathSciNet)
MR1415230

Zentralblatt MATH identifier
0885.60016

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 82C32: Neural nets [See also 68T05, 91E40, 92B20]

Keywords
Fluctuations Hopfield model overlap neural networks Laplace's method

Citation

Gentz, Barbara. A central limit theorem for the overlap in the Hopfield model. Ann. Probab. 24 (1996), no. 4, 1809--1841. doi:10.1214/aop/1041903207. https://projecteuclid.org/euclid.aop/1041903207


Export citation

References

  • [1] Amit, D. J., Gutfreund, H. and Sompolinsky, H. (1985). Spin-glass models of neural networks. Phy s. Rev. A 32 1007-1018.
  • [2] Bovier, A., Gay rard, V. and Picco, P. (1994). Gibbs states of the Hopfield model in the regime of perfect memory. Probab. Theory Related Fields 100, 329-363.
  • [3] Bovier, A. and Gay rard, V. (1996). An almost sure large deviation principle for the Hopfield model. Ann. Probab. 24 1444-1475.
  • [4] Bovier, A. and Gay rard, V. (1996). The retrieval phase of the Hopfield model: a rigorous analysis of the overlap distribution. Probab. Theory Related Fields. To appear.
  • [5] Ellis, R. S. (1985). Entropy, Large Deviations, and Statistical Mechanics. Springer, New York.
  • [6] Figotin, A. L. and Pastur, L. A. (1977). Exactly soluble model of a spin glass. Sov. J. Low Temp. Phy s. 3 378-383.
  • [7] Figotin, A. L. and Pastur, L. A. (1978). Theory of disordered spin sy stems. Theoret. and Math. Phy s. 35 403-414.
  • [8] Gentz, B. (1996). An almost sure central limit theorem for the overlap parameters in the Hopfield model. Stochast. Proc. Appl. 62 243-262.
  • [9] Hertz, J., Krogh, A. and Palmer, R. G. (1991). Introduction to the Theory of Neural Computation. Addison-Wesley, Reading, MA.
  • [10] Hopfield, J. J. (1982). Neural networks and physical sy stems with emergent collective computational abilities. Proc. Nat. Acad. Sci. U.S.A. 79 2554-2558.
  • [11] Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin.
  • [12] Martin-L ¨of, A. (1973). Mixing properties, differentiability of the free energy and the central limit theorem for a pure phase in the Ising model at low temperature. Comm. Math. Phy s. 32 75-92.
  • [13] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. IHES 81 73-205.