Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems

TL;DR

This paper introduces stabilized and scalable algorithms for entropy-regularized transport problems, improving convergence and efficiency for large-scale applications by combining multiple modifications and providing new theoretical insights.

Contribution

It presents a novel stabilized formulation with adaptive truncation and coarse-to-fine schemes, along with a new convergence analysis of the Sinkhorn algorithm for better understanding.

Findings

Enables solving larger problems with smaller regularization.

Achieves faster convergence and improved numerical stability.

Demonstrates effectiveness through numerical experiments.

Abstract

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 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 towards a better understanding of epsilon-scaling. Numerical examples illustrate efficiency and versatility of the modified algorithm.