Multimodal Sampling via Approximate Symmetries

TL;DR

This paper introduces a novel sampling method for multimodal distributions with approximate symmetries, leveraging a reference distribution and multilevel Monte Carlo techniques to improve efficiency in scientific computing.

Contribution

It proposes a new approach that constructs an exactly symmetric reference distribution from an approximately symmetric target, enhancing sampling efficiency.

Findings

The method accelerates sampling in multimodal distributions with approximate symmetries.

Numerical results on Ising models demonstrate improved efficiency over traditional methods.

The approach effectively bridges the gap between reference and target distributions using multilevel Monte Carlo.

Abstract

Sampling from multimodal distributions is a challenging task in scientific computing. When a distribution has an exact symmetry between the modes, direct jumps among them can accelerate the samplings significantly. However, the distributions from most applications do not have exact symmetries. This paper considers the distributions with approximate symmetries. We first construct an exactly symmetric reference distribution from the target one by averaging over the group orbit associated with the approximate symmetry. Next, we can apply the multilevel Monte Carlo methods by constructing a continuation path between the reference and target distributions. We discuss how to implement these steps with annealed importance sampling and tempered transitions. Compared with traditional multilevel methods, the proposed approach can be more effective since the reference and target distributions are much closer. Numerical results of the Ising models are presented to illustrate the efficiency of the proposed method.