Rigorous theory of approximate symmetry and a unified group-symmetric framework for general targets

Develop a rigorous theoretical characterization of the approximate symmetry phenomenon in target distributions—such as those arising in low-temperature statistical physics models—and construct a unified group-symmetric framework that applies to general target distributions lacking exact symmetry, thereby clarifying when and how group-based jumps can be systematically employed.

Background

The paper shows that group-induced averaging of Markov kernels can substantially improve mixing when the target distribution is invariant under a group action. It notes that in many practical settings exact invariance does not hold but an "approximate symmetry" often emerges, especially in low-temperature statistical physics models.

While such situations suggest the potential of group-based jumps, the authors point out that there is no general theoretical framework characterizing approximate symmetry or unifying its use for arbitrary targets without exact symmetry. Formalizing this would broaden the systematic applicability of group-averaged chains beyond the strictly symmetric case.