Safe Screening for Unbalanced Optimal Transport

TL;DR

This paper presents a novel Safe Screening framework tailored for the Unbalanced Optimal Transport problem, enabling faster optimization by efficiently identifying zero elements in sparse solutions without increasing complexity.

Contribution

It introduces a specialized Safe Screening approach for UOT with $\

Findings

Enhanced screening efficiency for UOT problems.

Effective application of Safe Screening to $\

Improved optimization speed without added complexity.

Abstract

This paper introduces a framework that utilizes the Safe Screening technique to accelerate the optimization process of the Unbalanced Optimal Transport (UOT) problem by proactively identifying and eliminating zero elements in the sparse solutions. We demonstrate the feasibility of applying Safe Screening to the UOT problem with $\ell_2$-penalty and KL-penalty by conducting an analysis of the solution's bounds and considering the local strong convexity of the dual problem. Considering the specific structural characteristics of the UOT in comparison to general Lasso problems on the index matrix, we specifically propose a novel approximate projection, an elliptical safe region construction, and a two-hyperplane relaxation method. These enhancements significantly improve the screening efficiency for the UOT's without altering the algorithm's complexity.