On the equivalence of Hopfield networks and Boltzmann Machines

Neural Networks, Volume 34, October 2012, Pages 1-9
Adriano Barra, Alberto Bernacchia, Enrica Santucci, Pierluigi Contucci

Dipartimento di Fisica, Sapienza Universita di Roma, Piazzale Aldo Moro 5, 00185, Roma, Italy

Department of Neurobiology, Yale University, 333 Cedar street, CT06510, New Haven, USA

Dipartimento di Matematica, Universita degli Studi dell’Aquila, Via Vetoio 1, 67010, L’Aquila, Italy

Dipartimento di Matematica, Alma Mater Studiorum Universita di Bologna, Piazza di Porta San Donato 5, 40126, Bologna, Italy

Abstract

A specific type of neural networks, the Restricted Boltzmann Machines (RBM), are implemented for classification and feature detection in machine learning. They are characterized by separate layers of visible and hidden units, which are able to learn efficiently a generative model of the observed data. We study a “hybrid” version of RBMs, in which hidden units are analog and visible units are binary, and we show that thermodynamics of visible units are equivalent to those of a Hopfield network, in which the N visible units are the neurons and the P hidden units are the learned patterns. We apply the method of stochastic stability to derive the thermodynamics of the model, by considering a formal extension of this technique to the case of multiple sets of stored patterns, which may act as a benchmark for the study of correlated sets.

Our results imply that simulating the dynamics of a Hopfield network, requiring the update of N neurons and the storage of N(N−1)/2 synapses, can be accomplished by a hybrid Boltzmann Machine, requiring the update of N+P neurons but the storage of only NP synapses. In addition, the well known glass transition of the Hopfield network has a counterpart in the Boltzmann Machine: it corresponds to an optimum criterion for selecting the relative sizes of the hidden and visible layers, resolving the trade-off between flexibility and generality of the model. The low storage phase of the Hopfield model corresponds to few hidden units and hence a overly constrained RBM, while the spin-glass phase (too many hidden units) corresponds to unconstrained RBM prone to overfitting of the observed data.

Go To Journal

 

Check Also

Light-Speed Encryption: Unlocking the Future with Spatially Incoherent Diffractive Neural Networks - Advances in Engineering

Light-Speed Encryption: Unlocking the Future with Spatially Incoherent Diffractive Neural Networks