Fast Optimal Transport through Sliced Wasserstein Generalized Geodesics

Understand

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake.

• In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based 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 in which the pivot measure is supported on a line.
• We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization.