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Conference Papers Year : 2020

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

Abstract

The problem of estimating Wasserstein distances between two densities living in high-dimension suffers from the curse of dimensionality: One needs an exponential (wrt dimension) number of samples to ensure that the distance between the two empirical measures is comparable to the distance between the original densities. Therefore, optimal transport (OT) geometry can only be used in machine learning if the OT problem is substantially regularized. On the other hand, one of the greatest achievements of the OT literature in recent years lies in regularity theory: Caffarelli showed that the OT map between two well behaved measures is Lipschitz, or, equivalently when considering 2-Wasserstein distances, that the Brenier convex potential (whose gradient yields an optimal map) is smooth. We propose in this work to draw inspiration from this theory and use regularity as a regularization tool. We give algorithms operating on two discrete measures that can recover nearly optimal transport maps with small distortion, or equivalently, nearly optimal Brenier potentials that are strongly convex and smooth. For univariate measures, we show that computing these potentials is equivalent to solving an isotonic regression problem with Lipschitz and strong monotonicity constraints. For multivariate measures the problem boils down to solving alternatively a convex QCQP and a discrete OT problem. We recover this way the values and gradients of the Brenier potential on sampled points, but also show more generally, that values and gradients of the potential can be computed out of sample, at the cost of solving a simpler QCQP for each evaluation. Building on these two formulations we propose algorithms to estimate and evaluate transport maps with desired regularity properties, benchmark their statistical performance, apply them to domain adaptation and visualize their action on a color transfer task.

Dates and versions

hal-02340371 , version 1 (30-10-2019)

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François-Pierre Paty, Alexandre d'Aspremont, Marco Cuturi. Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport. AISTATS 2020 - 23rd International Conference on Artificial Intelligence and Statistics, Jun 2020, Palermo / Virtual, Italy. ⟨hal-02340371⟩
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