Alternate minimization and doubly stochastic matrices

TL;DR

This paper studies Sinkhorn's algorithm for positive matrices, showing finite convergence properties for 2x2 matrices and characterizing their structure when convergence occurs quickly.

Contribution

It provides a detailed analysis of the convergence behavior of Sinkhorn's algorithm, especially for 2x2 matrices, and characterizes matrices with finite-step convergence.

Findings

Finite convergence occurs in at most two iterations for 2x2 matrices.

The structure of matrices with finite-step convergence is explicitly characterized.

Sinkhorn's algorithm reliably produces doubly stochastic matrices for positive matrices.

Abstract

Sinkhorn's alternative minimization algorithm applied to a positive $n\times n$ matrix converges to a doubly stochastic matrix. If the algorithm, applied to a $2\times 2$ matrix, converges in a finite number of iterations, then it converges in at most two iterations, and the structure of such matrices is determined.