Sampling from multi-modal distributions with polynomial query complexity in fixed dimension via reverse diffusion

TL;DR

This paper introduces a novel sampling algorithm for multi-modal distributions, including Gaussian mixtures, with polynomial query complexity in fixed dimensions, overcoming previous limitations like metastability and the need for prior mode knowledge.

Contribution

It presents the first polynomial-query complexity sampling method for broad distribution classes, utilizing reverse diffusion and self-normalized Monte Carlo estimators.

Findings

Handles all Gaussian mixtures without prior mode knowledge

Avoids metastability issues common in previous methods

Achieves polynomial query complexity in fixed dimensions

Abstract

Even in low dimensions, sampling from multi-modal distributions is challenging. We provide the first sampling algorithm for a broad class of distributions -- including all Gaussian mixtures -- with a query complexity that is polynomial in the parameters governing multi-modality, assuming fixed dimension. Our sampling algorithm simulates a time-reversed diffusion process, using a self-normalized Monte Carlo estimator of the intermediate score functions. Unlike previous works, it avoids metastability, requires no prior knowledge of the mode locations, and relaxes the well-known log-smoothness assumption which excluded general Gaussian mixtures so far.