Expressive Power and Approximation Errors of Restricted Boltzmann Machines

TL;DR

This paper analyzes the expressive power of Restricted Boltzmann Machines (RBMs) by characterizing the classes of distributions they can learn and providing bounds on their approximation errors based on the number of units.

Contribution

It introduces explicit classes of distributions learnable by RBMs and derives bounds on the Kullback-Leibler divergence, linking model size to approximation accuracy.

Findings

Bound on maximal Kullback-Leibler divergence for RBMs

Explicit classes of distributions learnable by RBMs

Guidelines for choosing hidden units to achieve desired approximation error

Abstract

We present explicit classes of probability distributions that can be learned by Restricted Boltzmann Machines (RBMs) depending on the number of units that they contain, and which are representative for the expressive power of the model. We use this to show that the maximal Kullback-Leibler divergence to the RBM model with $n$ visible and $m$ hidden units is bounded from above by $n - \left\lfloor \log(m+1) \right\rfloor - \frac{m+1}{2^{\left\lfloor\log(m+1)\right\rfloor}} \approx (n -1) - \log(m+1)$. In this way we can specify the number of hidden units that guarantees a sufficiently rich model containing different classes of distributions and respecting a given error tolerance.