A phase transition in sampling from Restricted Boltzmann Machines

• URL: http://arxiv.org/abs/2410.08423v1
• Date: Thu, 10 Oct 2024 23:51:22 GMT
• Title: A phase transition in sampling from Restricted Boltzmann Machines
• Authors: Youngwoo Kwon, Qian Qin, Guanyang Wang, Yuchen Wei,
• Abstract summary: We prove a phase transition phenomenon in the mixing time of the Gibbs sampler for a Restricted Boltzmann Machine. A key insight is the link between the Gibbs sampler and a dynamical system.
• Score: 2.6624014064407717
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $c_\star\approx-5.87$. 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.

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