Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems

Bernhard Schmitzer

Abstract: 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 $\varepsilon$-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 $\varepsilon$-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.