Results 1 
4 of
4
Hopfield models as generalized random mean field models. Mathematical aspects of spin glasses and neural networks
 3–89, Progr. Probab., 41 Birkhäuser
, 1998
"... Abstract: We give a comprehensive selfcontained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large devi ..."
Abstract

Cited by 30 (9 self)
 Add to MetaCart
Abstract: We give a comprehensive selfcontained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of “small α”; a particularly satisfactory result concerns a nontrivial regime of parameters in which we prove 1) the convergence of the local “mean fields ” to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) “propagation of chaos”, i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the “replica symmetric solution ” of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques.
An almost sure large deviation principle for the Hopfield model
 ANN. PROBAB 24
, 1995
"... We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic ‘overlap’parameters in the Hopfield model in the case where the number of random ‘patterns’, M, as a function of the system size N satisfies lim supM(N)/N = 0. In this case the r ..."
Abstract

Cited by 16 (6 self)
 Add to MetaCart
We prove a large deviation principle for the finite dimensional marginals of the Gibbs distribution of the macroscopic ‘overlap’parameters in the Hopfield model in the case where the number of random ‘patterns’, M, as a function of the system size N satisfies lim supM(N)/N = 0. In this case the rate function is independent of the disorder for almost all realizations of the patterns.
How glassy are neural networks?
, 2012
"... In this paper we continue our investigation on the high storage regime of a neural network with Gaussian patterns. Through an exact mapping between its partition function and one of a bipartite spin glass (whose parties consist of Ising and Gaussian spins respectively), we give a complete control of ..."
Abstract
 Add to MetaCart
In this paper we continue our investigation on the high storage regime of a neural network with Gaussian patterns. Through an exact mapping between its partition function and one of a bipartite spin glass (whose parties consist of Ising and Gaussian spins respectively), we give a complete control of the whole annealed region. The strategy explored is based on an interpolation between the bipartite system and two independent spin glasses built respectively by dichotomic and Gaussian spins: Critical line, behavior of the principal thermodynamic observables and their fluctuations as well as overlap fluctuations are obtained and discussed. Then, we move further, extending such an equivalence beyond the critical line, to explore the broken ergodicity phase under the assumption of replica symmetry and we show that the quenched free energy of this (analogical) Hopfield model can be described as a linear combination of the two quenched spinglass free energies even in the replica symmetric framework.
On the freeenergy of the Hopfield model
, 1995
"... The general theory of inhomogeneous meanfield systems of Ref. [18] provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model H fg N;p (S) = \Gamma 1 2N N X i;j=1 p X =1 i j S i S j for Ising spins S i and p random patterns = ( ..."
Abstract
 Add to MetaCart
The general theory of inhomogeneous meanfield systems of Ref. [18] provides a variational expression for the (almost sure) limiting free energy density of the Hopfield model H fg N;p (S) = \Gamma 1 2N N X i;j=1 p X =1 i j S i S j for Ising spins S i and p random patterns = ( 1 ; 2 ; \Delta \Delta \Delta ; N ) under the assumption that lim N!1 N \Gamma1 N X i=1 ffi i = ; i = ( 1 i ; 2 i ; \Delta \Delta \Delta ; p i ) exists (almost surely) in the space of probability measure over p copies of f\Gamma1; 1g. Including an "external field" term \Gamma P p =1 h P N i=1 i S i , we give a number of general properties of the freeenergy density and compute it for a): p = 2 in general, and b): p arbitrary when is uniform and at most the two components h 1 and h 2 are nonzero, obtaining the (almost sure) formula f(fi; h) = 1 2 f cw (fi; h 1 + h 2 ) + 1 2 f cw (fi; h 1 \Gamma h 2 ) for the free energy, where f cw deno...