A phase transition in sampling from Restricted Boltzmann Machines

Restricted Boltzmann Machines are a class of undirected graphical models that play a key role in deep learning and unsupervised learning. In this study, we prove a phase transition phenomenon in the mixing time of the Gibbs sampler for a one-parameter Restricted Boltzmann Machine. Specifically, the mixing time varies logarithmically, polynomially, and exponentially with the number of vertices depending on whether the parameter c is above, equal to, or below a critical value c_star≈-5.87. A key insight from our analysis is the link between the Gibbs sampler and a dynamical system, which we utilize to quantify the former based on the behavior of the latter. To study the critical case c= c_star, we develop a new isoperimetric inequality for the sampler's stationary distribution by showing that the distribution is nearly log-concave.