Tree-Sliced Wasserstein Distance on a System of Lines

Sliced Wasserstein (SW) distance in Optimal Transport (OT) is widely used in various applications thanks to its statistical effectiveness and computational efficiency. On the other hand, Tree Wassenstein (TW) and Tree-sliced Wassenstein (TSW) are instances of OT for probability measures where its ground cost is a tree metric. TSW also has a low computational complexity, i.e. linear to the number of edges in the tree. Especially, TSW is identical to SW when the tree is a chain. While SW is prone to loss of topological information of input measures due to relying on one-dimensional projection, TSW is more flexible and has a higher degree of freedom by choosing a tree rather than a line to alleviate the curse of dimensionality in SW. However, for practical applications, popular tree metric sampling methods are heavily built upon given supports, which limits their capacity to adapt to new supports. In this paper, we propose the Tree-Sliced Wasserstein distance on a System of Lines (TSW-SL), which brings a connection between SW and TSW. Compared to SW and TSW, our TSW-SL benefits from the higher degree of freedom of TSW while being suitable to dynamic settings as SW. In TSW-SL, we use a variant of the Radon Transform to project measures onto a system of lines, resulting in measures on a space with a tree metric, then leverage TW to efficiently compute distances between them. We empirically verify the advantages of TSW-SL over the traditional SW by conducting a variety of experiments on gradient flows, image style transfer, and generative models.