# Phase Diagram of Restricted Boltzmann Machines and Generalised Hopfield Networks with Arbitrary Priors

## TL;DR

This paper characterizes the phase diagram of Restricted Boltzmann Machines and generalized Hopfield networks with arbitrary priors, revealing the robustness and conditions for retrieval phases across different pattern and unit priors.

## Contribution

It provides a unified analysis of phase transitions in RBMs and Hopfield networks with arbitrary priors, extending understanding beyond Boolean patterns.

## Key findings

- Retrieval phase is robust across pattern distributions.
- Larger retrieval regions occur with peaked pattern entries and units.
- Retrieval exists at low temperature for all pattern priors.

## Abstract

Restricted Boltzmann Machines are described by the Gibbs measure of a bipartite spin glass, which in turn corresponds to the one of a generalised Hopfield network. This equivalence allows us to characterise the state of these systems in terms of retrieval capabilities, both at low and high load. We study the paramagnetic-spin glass and the spin glass-retrieval phase transitions, as the pattern (i.e. weight) distribution and spin (i.e. unit) priors vary smoothly from Gaussian real variables to Boolean discrete variables. Our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard Hopfield model with Boolean patterns. The retrieval region is larger when the pattern entries and retrieval units get more peaked and, conversely, when the hidden units acquire a broader prior and therefore have a stronger response to high fields. Moreover, at low load retrieval always exists below some critical temperature, for every pattern distribution ranging from the Boolean to the Gaussian case.