A topological insight into restricted Boltzmann machines

TL;DR

This paper explores the topological structure of Restricted Boltzmann Machines (RBMs), revealing their small-world nature, and demonstrates that constraining RBMs to a scale-free topology significantly reduces computational costs while maintaining or improving generative performance.

Contribution

It introduces a topological perspective on RBMs, showing that scale-free constraints can drastically reduce weights without sacrificing performance, and improves generative capabilities with more hidden neurons.

Findings

RBMs and GRBMs are bipartite graphs with small-world topology.

Scale-free constrained RBMs reduce weights by orders of magnitude with minimal performance loss.

Sparse models with more hidden neurons outperform standard RBMs in generative tasks.

Abstract

Restricted Boltzmann Machines (RBMs) and models derived from them have been successfully used as basic building blocks in deep artificial neural networks for automatic features extraction, unsupervised weights initialization, but also as density estimators. Thus, their generative and discriminative capabilities, but also their computational time are instrumental to a wide range of applications. Our main contribution is to look at RBMs from a topological perspective, bringing insights from network science. Firstly, here we show that RBMs and Gaussian RBMs (GRBMs) are bipartite graphs which naturally have a small-world topology. Secondly, we demonstrate both on synthetic and real-world datasets that by constraining RBMs and GRBMs to a scale-free topology (while still considering local neighborhoods and data distribution), we reduce the number of weights that need to be computed by a few orders of magnitude, at virtually no loss in generative performance. Thirdly, we show that, for a fixed number of weights, our proposed sparse models (which by design have a higher number of hidden neurons) achieve better generative capabilities than standard fully connected RBMs and GRBMs (which by design have a smaller number of hidden neurons), at no additional computational costs.