Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics

Wasserstein distance (WD) and the associated optimal transport plan have proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy for the squared WD, coined min-SWGG, which relies on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics with a pivot measure supported on a line. We notably provide a new closed form of the Wasserstein distance in the particular case where one of the distributions is supported on a line, allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to that of Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.