Title: Subspace Robust Wasserstein distances,
Abstract: Making sense of Wasserstein distances between discrete measures in
high-dimensional settings remains a challenge. Recent work has advocated a
two-step approach to improve robustness and facilitate the computation of
optimal transport, using for instance projections on random real lines, or a
preliminary quantization to reduce the number of points. We propose in this
work a new robust variant of the Wasserstein distance. This quantity captures
the maximal possible distance that can be realized between these two measures,
after they have been projected orthogonally on a lower k dimensional subspace.
We show that this distance inherits several favorably properties of OT, and
that computing it can be cast as a convex problem involving the top k
eigenvalues of the second order moment matrix of the displacements induced by a
transport plan. We provide algorithms to approximate the computation of this
saddle point using entropic regularization, and illustrate the interest of this
approach empirically.