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Optimal transport (OT), which provides a distance between two probabilitydistributions by considering their spatial locations, has been applied towidely diverse applications. Computing an OT problem requires solution oflinear programming with tight mass-conservation constraints. This requirementhinders its application to large-scale problems. To alleviate this issue, therecently proposed relaxed-OT approach uses a faster algorithm by relaxing suchconstraints. Its effectiveness for practical applications has beendemonstrated. Nevertheless, it still exhibits slow convergence. To this end,addressing a convex semi-relaxed OT, we propose a fast block-coordinateFrank-Wolfe (BCFW) algorithm, which gives sparse solutions. Specifically, weprovide their upper bounds of the worst convergence iterations, and equivalencebetween the linearization duality gap and the Lagrangian duality gap. Threefast variants of the proposed BCFW are also proposed. Numerical evaluations incolor transfer problem demonstrate that the proposed algorithms outperformstate-of-the-art algorithms across different settings.

Fast block-coordinate Frank-Wolfe algorithm for semi-relaxed optimal transport