Joint stochastic localization and applications

• URL: http://arxiv.org/abs/2505.13410v1
• Date: Mon, 19 May 2025 17:47:05 GMT
• Title: Joint stochastic localization and applications
• Authors: Tom Alberts, Yiming Xu, Qiang Ye,
• Abstract summary: We introduce a joint localization framework for constructing couplings between probability distributions.<n>We also introduce a family of distributional distances based on the couplings induced by joint localization.<n>The proposed distances also motivate new methods for distribution estimation that are of independent interest.
• Score: 7.22482471589943
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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