Regularity as Regularization: Smooth and Strongly Convex Brenier Potentials in Optimal Transport
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The problem of estimating Wasserstein distances in high-dimensional spaces suffers from the curse of dimensionality: Indeed, ones needs an exponential (w.r.t. dimension) number of samples for the distance between the two samples to be comparable to that between the two measures. Therefore, regularizing the optimal transport (OT) problem is crucial when using Wasserstein distances in machine learning. One of the greatest achievements of the OT literature in recent years lies in regularity theory: one can prove under suitable hypothesis that the OT map between two measures is Lipschitz, or, equivalently when studying 2-Wasserstein distances, that the Brenier convex potential (whose gradient yields an optimal map) is a smooth function. We propose in this work to go backwards, to adopt instead regularity as a regularization tool. We propose algorithms working on discrete measures that can recover nearly optimal transport maps that have small distortion, or, equivalently, nearly optimal Brenier potential that are strongly convex and smooth. For univariate measures, we show that computing these potentials is equivalent to solving an isotonic regression problem under Lipschitz and strong monotonicity constraints. For multivariate measures the problem boils down to a non-convex QCQP problem. We show that this QCQP can be lifted a semidefinite program. Most importantly, these potentials and their gradient can be evaluated on the measures themselves, but can more generally be evaluated on any new point by solving each time a QP. Building on these two formulations we propose practical algorithms to estimate and evaluate transport maps, and illustrate their performance statistically as well as visually on a color transfer task.
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