Abstract
We study Benamou’s domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove linear convergence of the algorithm with respect to the Kullback–Leibler divergence and illustrate the (potentially very slow) rates with numerical examples. On problems with sufficient geometric structure (such as Wasserstein distances between images) we expect much faster convergence. We then discuss important aspects of a computationally efficient implementation, such as adaptive sparsity, a coarse-to-fine scheme and parallelization, paving the way to numerically solving large-scale optimal transport problems. We demonstrate efficient numerical performance for computing the Wasserstein-2 distance between 2D images and observe that, even without parallelization, domain decomposition compares favorably to applying a single efficient implementation of the Sinkhorn algorithm in terms of runtime, memory and solution quality.
Highlights
1.1 Motivation(Computational) optimal transport Optimal transport is a fundamental optimization problem with applications in various branches of mathematics
Practical implementation and geometric large-scale examples In Sect. 6 we provide a practical version of the domain decomposition algorithm, leading to an efficient numerical method for large scale problems with geometric structure
Test data We focus on solving the Wasserstein-2 optimal transport problem, i.e. we set c(x, y) = x − y 2 but the scheme can be applied to arbitrary costs and we expect efficient performance for any cost of the form c(x, y) = h(x − y) for strictly convex h, such that the unique optimal plan is concentrated on the graph of a Monge map, see [11]
Summary
(Computational) optimal transport Optimal transport is a fundamental optimization problem with applications in various branches of mathematics. An introduction to computational optimal transport, an overview on available efficient algorithms, and applications can be found in [19]. Π (2) is obtained by optimizing π (1) on X{2,3} × Y , while keeping it fixed on X1 × Y In [2] it is shown that this algorithm converges to the globally optimal solution when μ is Lebesgue absolutely continuous and (X1, X2, X3) satisfy a ‘convex overlap’ condition, which roughly states that ‘if a function is convex on X{1,2} and X{2,3}, it must be convex on X ’. When the discretization is refined, it is shown that the optimal solution is recovered in the limit This requires an increasing number of discretization points in the three sets (X1, X2, X3) and numerically obtaining a good approximation of the globally optimal solution remains challenging with this method
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