Accelerating the Sinkhorn-Knopp iteration by Arnoldi-type methods

TL;DR

This paper introduces Arnoldi-type methods to speed up the Sinkhorn-Knopp iteration for matrix balancing, by reformulating the problem as a nonlinear eigenvalue problem and improving convergence especially with clustered eigenvalues.

Contribution

It proposes novel Arnoldi-based algorithms for faster convergence in matrix balancing, extending the classical Sinkhorn-Knopp iteration through an eigenvalue problem reformulation.

Findings

Significant acceleration of convergence for clustered eigenvalues

Effective adaptation of power and Arnoldi methods for matrix balancing

Numerical results demonstrate improved efficiency over traditional methods

Abstract

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.