  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.
