Accelerating operator Sinkhorn iteration with overrelaxation

TL;DR

This paper introduces accelerated operator Sinkhorn iterations with overrelaxation, providing convergence analysis and demonstrating improved performance in applications.

Contribution

It develops and analyzes accelerated Sinkhorn methods with overrelaxation, extending previous matrix scaling results to operator scaling.

Findings

Accelerated methods converge faster than standard Sinkhorn iteration.

Optimal relaxation parameters are derived from Young's SOR theorem.

Numerical experiments show improved performance in applications.

Abstract

We propose accelerated versions of the operator Sinkhorn iteration for operator scaling using successive overrelaxation. We analyze the local convergence rates of these accelerated methods via linearization, which allows us to determine the asymptotically optimal relaxation parameter based on Young's SOR theorem. Using the Hilbert metric on positive definite cones, we also obtain a global convergence result for a geodesic version of overrelaxation in a specific range of relaxation parameters. These techniques generalize corresponding results obtained for matrix scaling by Thibault et al. (Algorithms, 14(5):143, 2021) and Lehmann et al. (Optim. Lett., 16(8):2209--2220, 2022). Numerical experiments demonstrate that the proposed methods outperform the original operator Sinkhorn iteration in certain applications.