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Article Dans Une Revue SIAM Journal on Mathematics of Data Science Année : 2022

Accelerated Bregman primal-dual methods applied to optimal transport and Wasserstein Barycenter problems

Résumé

This paper discusses the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman's kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.
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Dates et versions

hal-03806188 , version 1 (07-10-2022)

Identifiants

Citer

Antonin Chambolle, Juan Pablo Contreras. Accelerated Bregman primal-dual methods applied to optimal transport and Wasserstein Barycenter problems. SIAM Journal on Mathematics of Data Science, 2022, 4 (4), pp.1369-1395. ⟨10.1137/22M1481865⟩. ⟨hal-03806188⟩
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