Numerically stable variants of overrelaxation for operator Sinkhorn iteration

TL;DR

This paper develops numerically stable variants of overrelaxation methods for the operator Sinkhorn iteration, enabling more accurate and stable solutions for scaling problems in ill-conditioned cases.

Contribution

It introduces stable versions of SOR-accelerated OSI that maintain numerical stability while leveraging preconditioning effects.

Findings

Stable SOR-accelerated OSI works in ill-conditioned cases.

Numerical experiments confirm improved stability and accuracy.

Modified algorithms outperform direct SOR implementations.

Abstract

We consider accelerated versions of the operator Sinkhorn iteration (OSI) for solving scaling problems for completely positive maps. Based on the interpretation of OSI as alternating fixed point iteration, it has been recently proposed to achieve acceleration by means of nonlinear successive overrelaxation (SOR), e.g.~with respect to geodesics in Hilbert metric. The direct implementation of the proposed SOR algorithms, however, can be numerically unstable for ill-conditioned instances, limiting the achievable accuracy. Here we derive equivalent versions of OSI with SOR where, similar to the original OSI formulation, scalings are applied on the fly in order to take advantage of preconditioning effects. Numerical experiments confirm that this modification allows for numerically stable SOR-acceleration of OSI even in ill-conditioned cases.