Joint stochastic localization and applications

Abstract

Stochastic localization is a pathwise analysis technique that has emerged as a powerful tool in high-dimensional probability and sampling. In this work, we extend stochastic localization to a joint framework for coupling probability measures and explore its applications in distributional data analysis. We first unify existing stochastic localization processes under Eldan's $\alpha$-scheme and characterize their localization rates. Building on this, we introduce a joint scheme to couple probability measures via concurrent $\alpha$-schemes driven by a shared Brownian motion. This construction is canonical and induces a family of metrics on the space of probability measures, which we call Eldan's $\alpha$-distance. Alternative variants that extrapolate optimal Gaussian couplings to log-concave measures are also discussed. We study the theoretical properties of Eldan's $\alpha$-distance, including its restriction to Gaussian measures and its behavior under affine transformations. For $\alpha = 0$, we show it is topologically equivalent to the $2$-Wasserstein distance for measures supported on a common compact set; we also relate its weighted variants to linearized optimal transport in Wiener space and to score-matching objectives in training diffusion models. Computationally, we develop efficient estimators for Eldan's $\alpha$-distance in the cases $\alpha=0$ and $\alpha=1/2$, with rigorous error guarantees for log-concave and finitely supported measures in the former setting and Gaussian measures in the latter. Finally, we apply Eldan's $\alpha$-distance as a scalable surrogate for the $2$-Wasserstein distance to enable fast pairwise distance estimation and approximate computation of Wasserstein barycenters.