# A phase transition in sampling from Restricted Boltzmann Machines

arXiv: https://arxiv.org/abs/2410.08423 · HTML: https://mlsanity.com/paper/2410.08423/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 33 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 5 of 5)

- [Log-concave sampling: Metropolis-Hastings algorithms are fast](https://mlsanity.com/paper/1801.02309/) (2018)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [Dimension-free Mixing for High-dimensional Bayesian Variable Selection](https://mlsanity.com/paper/2105.05719/) (2021)
- [On Mixing Rates for Bayesian CART](https://mlsanity.com/paper/2306.00126/) (2023)
- [Dimension-free Relaxation Times of Informed MCMC Samplers on Discrete Spaces](https://mlsanity.com/paper/2404.03867/) (2024)

## Built on by (most cited first, 1 of 1)

- [Rapid mixing in positively weighted restricted Boltzmann machines](https://mlsanity.com/paper/2604.00963/) (2026)

## Abstract (arXiv)

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.