Block-coordinate Frank-Wolfe algorithm and convergence analysis for semi-relaxed optimal transport problem

TL;DR

This paper introduces a fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport, providing convergence analysis and demonstrating superior performance in applications like color transfer.

Contribution

It proposes a novel BCFW algorithm for semi-relaxed OT with proven convergence bounds and fast variants, improving computational efficiency over existing methods.

Findings

Algorithms outperform state-of-the-art in color transfer tasks

Convergence bounds are established for the proposed methods

Fast variants significantly reduce computation time

Abstract

The optimal transport (OT) problem has been used widely for machine learning. It is necessary for computation of an OT problem to solve linear programming with tight mass-conservation constraints. These constraints prevent its application to large-scale problems. To address this issue, loosening such constraints enables us to propose the relaxed-OT method using a faster algorithm. This approach has demonstrated its effectiveness for applications. However, it remains slow. As a superior alternative, we propose a fast block-coordinate Frank-Wolfe (BCFW) algorithm for a convex semi-relaxed OT. Specifically, we prove their upper bounds of the worst convergence iterations, and equivalence between the linearization duality gap and the Lagrangian duality gap. Additionally, we develop two fast variants of the proposed BCFW. Numerical experiments have demonstrated that our proposed algorithms are effective for color transfer and surpass state-of-the-art algorithms. This report presents a short version of arXiv:2103.05857.