Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems
Schmitzer, Bernhard (2016), Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. https://basepub.dauphine.fr/handle/123456789/17199
TypeDocument de travail / Working paper
External document linkhttps://arxiv.org/abs/1610.06519
Series titleCahier de recherche CEREMADE, Université Paris-Dauphine
MetadataShow full item record
Abstract (EN)Scaling algorithms for entropic transport-type problems have become a very popular numerical method, encompassing Wasserstein barycenters, multi-marginal 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 propose 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 allows to solve larger problems with smaller regularization and negligible truncation error. A new convergence analysis of the Sinkhorn algorithm is developed, working towards a better understanding of epsilon-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.
Subjects / KeywordsMulti-Scale; Sinkhorn algorithm; Sparsity; Optimal Transport
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