Restricted Boltzmann Machines as Models of Interacting Variables

TL;DR

This paper investigates how the choice of activation functions in Restricted Boltzmann Machines influences the types of distributions they can model, providing exact expressions for the induced interactions and analyzing their properties.

Contribution

It derives exact marginal distributions for RBMs with various activation functions and examines how these affect the model's ability to approximate binary data distributions.

Findings

Weak parameters lead to similar interaction patterns across activation functions

RBMs trained on MNIST are well-approximated by low-order interaction models

The activation function choice impacts the complexity of the modeled distributions

Abstract

We study the type of distributions that Restricted Boltzmann Machines (RBMs) with different activation functions can express by investigating the effect of the activation function of the hidden nodes on the marginal distribution they impose on observed binary nodes. We report an exact expression for these marginals in the form of a model of interacting binary variables with the explicit form of the interactions depending on the hidden node activation function. We study the properties of these interactions in detail and evaluate how the accuracy with which the RBM approximates distributions over binary variables depends on the hidden node activation function and on the number of hidden nodes. When the inferred RBM parameters are weak, an intuitive pattern is found for the expression of the interaction terms which reduces substantially the differences across activation functions. We show that the weak parameter approximation is a good approximation for different RBMs trained on the MNIST dataset. Interestingly, in these cases, the mapping reveals that the inferred models are essentially low order interaction models.