Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 41, Issue 3, Pages A1443-A1481Publisher
SIAM PUBLICATIONS
DOI: 10.1137/16M1106018
Keywords
optimal transport; convex optimization; entropy regularization; multiscale
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Funding
- European Research Council through project SIGMA-Vision
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Scaling algorithms for entropic transport-type problems have become a very popular numerical method, encompassing Wasserstein barycenters, multimarginal problems, gradient flows, and unbalanced transport. However, a standard implementation of the scaling algorithm has several numerical limitations: the scaling factors diverge and convergence becomes impractically slow as the entropy regularization approaches zero. Moreover, handling the dense kernel matrix becomes unfeasible for large problems. To address this, we combine several modifications: A log-domain stabilized formulation, the well-known epsilon-scaling heuristic, an adaptive truncation of the kernel, and a coarse-to-fine scheme. This permits the solution of larger problems with smaller regularization and negligible truncation error. A new convergence analysis of the Sinkhorn algorithm is developed, working toward a better understanding of epsilon-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.
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