Sinkhorn limits in finitely many steps

TL;DR

This paper proves that for nonnegative matrices with a nonzero sigma-diagonal, finite-step convergence of the Sinkhorn scaling sequence implies convergence within at most two steps, simplifying the understanding of the scaling process.

Contribution

It establishes a new theoretical result that finite convergence in Sinkhorn scaling occurs in at most two steps for matrices with a nonzero sigma-diagonal.

Findings

Finite convergence implies convergence within two steps.

Convergence occurs after at most two scalings for certain matrices.

Provides theoretical insight into Sinkhorn scaling behavior.

Abstract

Applied to a nonnegative $m\times n$ matrix with a nonzero $\sigma$-diagonal, the sequence of matrices constructed by alternate row and column scaling conveges to a doubly stochastic matrix. It is proved that if this sequence converges after only a finite number of scalings, then it converges after at most two scalings.