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Classification and lumpability in the stochastic Hopfield model

Part of: Distribution theory Artificial intelligence (68Txx) Special processes

Published online by Cambridge University Press:  19 February 2016

R. L. Paige
R. L. Paige*
Affiliation:
Texas Technological University
*
∗ Postal address: Department of Mathematics and Statistics, Texas Technological University, Lubbock, TX 79409, USA. Email address: rpaige@math.ttu.edu
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Abstract

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

Keywords

Classification timesneural networks

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 62E17: Approximations to distributions (nonasymptotic) 68T01: General
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 4 , December 2001 , pp. 930 - 943
Copyright
Copyright © Applied Probability Trust 2001 

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References

Amit, D. J. (1989). Modeling Brain Function. Cambridge University Press.CrossRefGoogle Scholar
Amit, D. J., Gutfreund, H. and Sompolinsky, H. (1985). Spin-glass models for neural networks. Phys. Rev. A,32, 1007–1018.CrossRefGoogle ScholarPubMed
Butler, R. W. (2000). Reliabilities for feedback systems and their saddlepoint approximation. Statist. Sci. 15, 279–298.CrossRefGoogle Scholar
Daniels, H. (1954). Saddlepoint approximations in statistics. Ann. Math. Statist. 25, 631–650.CrossRefGoogle Scholar
Daniels, H. (1987). Tail probability approximations. Internat. Statist. Rev. 55, 37–48.CrossRefGoogle Scholar
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach Intellig. 6, 720–741.Google ScholarPubMed
Hertz, J. A., Krogh, A. S. and Palmer, R. G. (1991). Introduction to the Theory of Neural Computation. Addison-Wesley, New York.Google Scholar
Hoffman, R. E. (1992). Attractor neural networks and psychotic disorders. Psych. Ann. 22, 119–124.CrossRefGoogle Scholar
Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proc. Nat. Acad. Sci. USA 79, 2554–2558.CrossRefGoogle ScholarPubMed
Kam, M. and Cheng, R. (1989). Decision-making with the Boltzmann machine. In Proc. Amer. Control Conf. (Pittsburgh, PA, June 1989) Vol. 1, pp. 902–907.Google Scholar
Kemeny, J. and Snell, L. (1969). Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
Kleinfield, D. and Somopolinsky, H. (1989). Associative neural models for central pattern generators. In Methods in Neuronal Modeling: From Synapses to Networks, eds Koch, C. and Segev, I., MIT Press, Cambridge, MA, pp. 195–246.Google Scholar
Sumita, U. and Rieders, M. (1988). First passage times and lumpability of semi-Markov processes. J. Appl. Prob. 25, 675–687.CrossRefGoogle Scholar
Whittle, P. (1991). Neural nets and implicit inference. Ann. Appl. Prob. 1, 173–188.CrossRefGoogle Scholar