  Optimal transport (OT), which provides a distance between two probability
distributions by considering their spatial locations, has been applied to
widely diverse applications. Computing an OT problem requires solution of
linear programming with tight mass-conservation constraints. This requirement
hinders its application to large-scale problems. To alleviate this issue, the
recently proposed relaxed-OT approach uses a faster algorithm by relaxing such
constraints. Its effectiveness for practical applications has been
demonstrated. Nevertheless, it still exhibits slow convergence. To this end,
addressing a convex semi-relaxed OT, we propose a fast block-coordinate
Frank-Wolfe (BCFW) algorithm, which gives sparse solutions. Specifically, we
provide their upper bounds of the worst convergence iterations, and equivalence
between the linearization duality gap and the Lagrangian duality gap. Three
fast variants of the proposed BCFW are also proposed. Numerical evaluations in
color transfer problem demonstrate that the proposed algorithms outperform
state-of-the-art algorithms across different settings.
