Joint stochastic localization and applications

TL;DR

This paper develops a unified framework for stochastic localization, extending optimal couplings and introducing new distributional distances, with applications in normal approximation and distribution estimation.

Contribution

It unifies existing stochastic localization schemes, introduces a joint framework for couplings, and proposes new distributional distances with practical applications.

Findings

Extended optimal couplings to log-concave distributions.

Introduced a new distributional distance equivalent to 2-Wasserstein on compact sets.

Proposed new methods for distribution estimation.

Abstract

Stochastic localization is a pathwise analysis technique originating from convex geometry. This paper explores certain algorithmic aspects of stochastic localization as a computational tool. First, we unify various existing stochastic localization schemes and discuss their localization rates and regularization. We then introduce a joint stochastic localization framework for constructing couplings between probability distributions. As an initial application, we extend the optimal couplings between normal distributions under the 2-Wasserstein distance to log-concave distributions and derive a normal approximation result. As a further application, we introduce a family of distributional distances based on the couplings induced by joint stochastic localization. Under a specific choice of the localization process, the induced distance is topologically equivalent to the 2-Wasserstein distance for probability measures supported on a common compact set. Moreover, weighted versions of this distance are related to several statistical divergences commonly used in practice. The proposed distances also motivate new methods for distribution estimation that are of independent interest.