Closed Form of a Generalized Sinkhorn Limit

TL;DR

This paper derives a closed-form solution for the generalized Sinkhorn limit of 2x2 matrices and discusses extensions, showing that entries in the limit are algebraic over input data with bounded degree.

Contribution

It provides the first explicit closed form for the generalized Sinkhorn limit of 2x2 matrices and establishes algebraic properties of the limit entries for larger matrices.

Findings

Closed form for 2x2 generalized Sinkhorn limit derived.

Entries in the limit are algebraic over input data.

Degree of algebraicity bounded by a combinatorial expression.

Abstract

The Kruithof iterative scaling process, which adjusts matrices to meet target row and column sums, is a longstanding problem that lacks a general closed form for its limit. While Nathanson derived the closed form for the Sinkhorn limit of $2\times 2$ matrices when target row and column sums are 1, and recent work by Rowland and Wu has advanced understanding of Sinkhorn limits for $3\times 3$, and general $n\times m$ matrices through polynomials, a "generalized Sinkhorn limit" (i.e. the original "Kruithof limit", with arbitrary target sums) remains elusive. Here, we derive the closed form for the generalized Sinkhorn limit of $2\times 2$ matrices, and discuss how this approach can be extended to larger matrices. More significantly, we prove that for any positive $n \times m$ matrix and positive target row and column sums, each entry in the generalized Sinkhorn limit is algebraic over the input data with degree at most $\binom{n+m-2}{n-1}$.