On the equivalence of Hopfield Networks and Boltzmann Machines

TL;DR

This paper demonstrates the thermodynamic equivalence between a hybrid Boltzmann Machine with analog hidden units and binary visible units and a Hopfield network, providing insights into their dynamics, storage, and phase transitions.

Contribution

It establishes a formal equivalence between Hopfield networks and a hybrid Boltzmann Machine, offering a new perspective on their thermodynamics and capacity trade-offs.

Findings

Thermodynamics of the hybrid RBM are equivalent to those of a Hopfield network.

Simulating a Hopfield network can be achieved with fewer synapses using a hybrid Boltzmann Machine.

The glass transition in Hopfield networks relates to the optimal size ratio of hidden to visible units.

Abstract

A specific type of neural network, the Restricted Boltzmann Machine (RBM), is implemented for classification and feature detection in machine learning. RBM is 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 RBM's, 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.