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A phase transition in sampling from restricted Boltzmann machines

Y.-J. Kwon et al.

Annals of Applied Probability2026https://doi.org/10.1214/25-aap2265article
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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⋆≈−5.87. A key insight from our analysis is the link between the Gibbs sampler and a dynamic system, which we use to quantify the former based on the behavior of the latter. To study the critical case c=c⋆, we develop a new isoperimetric inequality for the sampler’s stationary distribution by showing that the distribution is nearly log-concave. Conditions for rapid and torpid convergence of a class of generic binomial chain are also provided.

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https://doi.org/https://doi.org/10.1214/25-aap2265

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@article{y.-j.2026,
  title        = {{A phase transition in sampling from restricted Boltzmann machines}},
  author       = {Y.-J. Kwon et al.},
  journal      = {Annals of Applied Probability},
  year         = {2026},
  doi          = {https://doi.org/https://doi.org/10.1214/25-aap2265},
}

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A phase transition in sampling from restricted Boltzmann machines

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