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 $α$-scheme and characterize their localization rates. Building on this, we introduce a joint scheme to couple probability measures via concurrent $α$-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 $α$-distance. Alternative variants that extrapolate optimal Gaussian couplings to log-concave measures are also discussed. We study the theoretical properties of Eldan's $α$-distance, including its restriction to Gaussian measures and its behavior under affine transformations. For $α= 0$, 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 $α$-distance in the cases $α=0$ and $α=1/2$, 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 $α$-distance as a scalable surrogate for the $2$-Wasserstein distance to enable fast pairwise distance estimation and approximate computation of Wasserstein barycenters.

翻译：随机定位是一种路径分析方法，已成为高维概率与抽样领域的强大工具。本文将其扩展为耦合概率测度的联合框架，并探讨其在分布数据分析中的应用。首先，我们统一了Eldan的α方案下的现有随机定位过程，并刻画了其定位速率。在此基础上，我们提出了一种联合方案，通过由共同布朗运动驱动的并发α方案来耦合概率测度。这一构造是典范的，并在概率测度空间上导出度量族，称为Eldan的α-距离。我们还讨论了将最优高斯耦合外推至对数凹测度的替代变体。我们研究了Eldan的α-距离的理论性质，包括它在高斯测度上的限制以及仿射变换下的行为。当α=0时，我们证明对于支撑于共同紧集上的测度，它拓扑等价于2- Wasserstein距离；此外，我们将其加权变体与Wiener空间中的线性化最优输运以及训练扩散模型中的得分匹配目标联系起来。在计算方面，我们针对α=0和α=1/2的情形开发了Eldan的α-距离的高效估计器，前者对于对数凹测度和有限支撑测度具有严格的误差保证，后者则针对高斯测度。最后，我们将Eldan的α-距离作为2- Wasserstein距离的可扩展替代，用于实现快速成对距离估计以及Wasserstein重心近似计算。