Wasserstein gradient flow for optimal probability measure decomposition

TL;DR

This paper introduces Wasserstein gradient flow algorithms to decompose probability measures into sub-measures, optimizing clustering-related loss functions, with theoretical analysis and numerical validation.

Contribution

It presents a novel approach using Wasserstein gradient flow for measure decomposition, including convergence analysis and practical algorithms.

Findings

Algorithms converge reliably in experiments.

Decomposition effectively minimizes clustering loss functions.

Numerical results validate theoretical insights.

Abstract

We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We analytically explore the structures of the support of optimal sub-measures and introduce algorithms based on Wasserstein gradient flow, demonstrating their convergence. Numerical results illustrate the implementability of our algorithms and provide further insights.