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Estimating Wasserstein distances between two high-dimensional densitiessuffers from the curse of dimensionality: one needs an exponential (wrtdimension) number of samples to ensure that the distance between two empiricalmeasures is comparable to the distance between the original densities.Therefore, optimal transport (OT) can only be used in machine learning if it issubstantially regularized. On the other hand, one of the greatest achievementsof the OT literature in recent years lies in regularity theory: Caffarellishowed that the OT map between two well behaved measures is Lipschitz, orequivalently when considering 2-Wasserstein distances, that Brenier convexpotentials (whose gradient yields an optimal map) are smooth. We propose inthis work to draw inspiration from this theory and use regularity as aregularization tool. We give algorithms operating on two discrete measures thatcan recover nearly optimal transport maps with small distortion, orequivalently, nearly optimal Brenier potentials that are strongly convex andsmooth. The problem boils down to solving alternatively a convex QCQP and adiscrete OT problem, granting access to the values and gradients of the Brenierpotential not only on sampled points, but also out of sample at the cost ofsolving a simpler QCQP for each evaluation. We propose algorithms to estimateand evaluate transport maps with desired regularity properties, benchmark theirstatistical performance, apply them to domain adaptation and visualize theiraction on a color transfer task.

Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport