  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 of the measures to reduce the size of their support.
We propose in this work a "max-min" robust variant of the Wasserstein distance
by considering the maximal possible distance that can be realized between two
measures, assuming they can be projected orthogonally on a lower
$k$-dimensional subspace. Alternatively, we show that the corresponding
"min-max" OT problem has a tight convex relaxation which can be cast as that of
finding an optimal transport plan with a low transportation cost, where the
cost is alternatively defined as the sum of the $k$ largest eigenvalues of the
second order moment matrix of the displacements (or matchings) corresponding to
that plan (the usual OT definition only considers the trace of that matrix). We
show that both quantities inherit several favorable properties from the OT
geometry. We propose two algorithms to compute the latter formulation using
entropic regularization, and illustrate the interest of this approach
empirically.
