A Sparse Multiscale Algorithm for Dense Optimal Transport

TLDR: This work provides a framework to verify global optimality of a discrete transport plan locally and explicitly describes how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance.

Abstract: Discrete optimal transport solvers do not scale well on dense large problems since they do not explicitly exploit the geometric structure of the cost function. In analogy to continuous optimal transport, we provide a framework to verify global optimality of a discrete transport plan locally. This allows the construction of an algorithm to solve large dense problems by considering a sequence of sparse problems instead. The algorithm lends itself to being combined with a hierarchical multiscale scheme. Any existing discrete solver can be used as internal black-box. We explicitly describe how to select the sparse sub-problems for several cost functions, including the noisy squared Euclidean distance. Significant reductions in run-time and memory requirements have been observed.