Weak Poincaré Inequalities and Mixing from Non-Worst-Case Initializations

Amit Rajaraman (MIT) https://simons.berkeley.edu/talks/amit-rajaraman-mit-2026-09-25 Local to Global Principles There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincaré inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. Under such conditions, a Poincaré inequality does not hold, necessitating new tools to prove sampling guarantees. We develop a framework to analyze such initializations, based on establishing so-called "weak Poincaré inequalities". As an application, we prove that "simulated annealing" samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. In this talk, we will focus on an application of our techniques to sampling from mixtures of log-concave distributions using data-based initializations. Based on joint work with Brice Huang, Sidhanth Mohanty, and David X Wu. Available at https://arxiv.org/abs/2411.09075.