Subspace Robust Wasserstein distances
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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.
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