François-Pierre Paty

François-Pierre Paty, Philippe Choné, Francis Kramarz

The theory of weak optimal transport (WOT), introduced by [Gozlan et al., 2017], generalizes the classic Monge-Kantorovich framework by allowing the transport cost between one point and the points it is matched with to be nonlinear. In the so-called barycentric version of WOT, the cost for transporting a point $x$ only depends on $x$ and on the barycenter of the points it is matched with. This aggregation property of WOT is appealing in machine learning, economics and finance. Yet algorithms to compute WOT have only been developed for the special case of quadratic barycentric WOT, or depend on neural networks with no guarantee on the computed value and matching. The main difficulty lies in the transportation constraints which are costly to project onto. In this paper, we propose to use mirror descent algorithms to solve the primal and dual versions of the WOT problem. We also apply our algorithms to the variant of WOT introduced by [Choné et al., 2022] where mass is distributed from one space to another through unnormalized kernels (WOTUK). We empirically compare the solutions of WOT and WOTUK with classical OT. We illustrate our numerical methods to the economic framework of [Choné and Kramarz, 2021], namely the matching between workers and firms on labor markets.

François-Pierre Paty, Marco Cuturi

Regularizing the optimal transport (OT) problem has proven crucial for OT theory to impact the field of machine learning. For instance, it is known that regularizing OT problems with entropy leads to faster computations and better differentiation using the Sinkhorn algorithm, as well as better sample complexity bounds than classic OT. In this work we depart from this practical perspective and propose a new interpretation of regularization as a robust mechanism, and show using Fenchel duality that any convex regularization of OT can be interpreted as ground cost adversarial. This incidentally gives access to a robust dissimilarity measure on the ground space, which can in turn be used in other applications. We propose algorithms to compute this robust cost, and illustrate the interest of this approach empirically.

François-Pierre Paty, Alexandre d'Aspremont, Marco Cuturi

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transport (OT) can only be used in machine learning if it 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 Brenier convex potentials (whose gradient yields an optimal map) are 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. The problem boils down to solving alternatively a convex QCQP and a discrete OT problem, granting access to the values and gradients of the Brenier potential not only on sampled points, but also out of sample at the cost of solving a simpler QCQP for each evaluation. 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.

François-Pierre Paty, Marco Cuturi

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.