Exact training of Restricted Boltzmann machines on intrinsically low dimensional data

TL;DR

This paper provides an exact analysis of Restricted Boltzmann machines trained on low-dimensional data, revealing fundamental issues with standard training methods and proposing a convex relaxation approach for better solutions.

Contribution

It offers an exact treatment for RBMs on low-dimensional data, clarifies why standard training fails, and introduces a convex relaxation for improved optimization.

Findings

Standard training fails due to phase transitions

Exact solutions are obtained for 1D and 2D cases

Convex relaxation yields unique solutions

Abstract

The restricted Boltzmann machine is a basic machine learning tool able, in principle, to model the distribution of some arbitrary dataset. Its standard training procedure appears however delicate and obscure in many respects. We bring some new insights to it by considering the situation where the data have low intrinsic dimension, offering the possibility of an exact treatment and revealing a fundamental failure of the standard training procedure. The reasons for this failure \textemdash~like the occurrence of first-order phase transitions during training~\textemdash \ are clarified thanks to a Coulomb interactions reformulation of the model. In addition a convex relaxation of the original optimization problem is formulated thereby resulting in a unique solution, obtained in precise numerical form on $d=1,2$ study cases, while a constrained linear regression solution can be conjectured on the basis of an information theory argument.