Probabilitas Seminar: June Vuong

The Probabilitas Seminar series focuses on high-dimensional problems that combine statistics, probability, information theory, computer science, and other related fields. The upcoming seminar takes place on Friday, March 7, from 10:30-11:30am EST. The talk will be hybrid, both in-person in Science Center 316 and on Zoom (please contact the department for Zoom information). This week's guest will be June Vuong of the Miller Institute at UC Berkeley.

Efficiently learning and sampling from multimodal distributions using data-based initialization

We consider the problem of sampling a multimodal distribution with a Markov chain given a small number of samples from the stationary measure. Although mixing can be arbitrarily slow, we show that if the Markov chain has a kth order spectral gap, initialization from a set of O~(k/ε2) samples from the stationary distribution will, with high probability over the samples, efficiently generate a sample whose conditional law is ε-close in TV distance to the stationary measure. In particular, this applies to mixtures of k distributions satisfying a Poincaré inequality, with faster convergence when they satisfy a log-Sobolev inequality. Our bounds are stable to perturbations to the Markov chain, and in particular work for Langevin diffusion over Rd with score estimation error, as well as Glauber dynamics combined with approximation error from pseudolikelihood estimation. This justifies the success of data-based initialization for score matching methods despite slow mixing for the data distribution, and improves and generalizes the results of Koehler and Vuong (2023) to have linear, rather than exponential, dependence on k and apply to arbitrary semigroups. As a consequence of our results, we show for the first time that a natural class of low-complexity Ising measures can be efficiently learned from samples.

Based on joint work with Frederic Koehler and Holden Lee.