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The entries of the Sinkhorn limit of an $m \times n$ matrix

Eric Rowland, Jason Wu

TL;DR

This work investigates exact algebraic descriptions of Sinkhorn limits, focusing on the top-left entry of the Sinkhorn limit for positive $m \times n$ matrices. By combining Gröbner bases for small cases, PSLQ-based interpolation for larger cases, and symmetry-driven reformulations via class sums, the authors conjecture a universal polynomial relation of degree $\binom{2n-2}{n-1}$ for square matrices and extend the framework to rectangular and Kruithof limits. They reveal a rich combinatorial structure in the coefficients, expressed through minors, determinant-like constructions, and equivalence classes, and propose a determinant-based architecture that may decompose across connected components. The results aim to provide exact, algebraic handles on Sinkhorn limits with potential implications for numerical stability, combinatorial matrix theory, and generalized scaling problems.

Abstract

We use a variety of computational tools to obtain a degree-$\binom{m + n - 2}{m - 1}$ polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive $m \times n$ matrix. The degree of this equation has a combinatorial interpretation as the number of minors of an $(m - 1) \times (n - 1)$ matrix, and the coefficients involve a determinant formula that reflects new combinatorial structure on sets of minor specifications. The tools we use include Gröbner bases, which produce equations for small matrices; the PSLQ algorithm, which produces equations for larger matrices as part of an interpolation effort that required 1.5 years of CPU time; and ChatGPT o3-mini-high, which identified the signs of the off-diagonal entries in the determinant formula.

The entries of the Sinkhorn limit of an $m \times n$ matrix

TL;DR

This work investigates exact algebraic descriptions of Sinkhorn limits, focusing on the top-left entry of the Sinkhorn limit for positive matrices. By combining Gröbner bases for small cases, PSLQ-based interpolation for larger cases, and symmetry-driven reformulations via class sums, the authors conjecture a universal polynomial relation of degree for square matrices and extend the framework to rectangular and Kruithof limits. They reveal a rich combinatorial structure in the coefficients, expressed through minors, determinant-like constructions, and equivalence classes, and propose a determinant-based architecture that may decompose across connected components. The results aim to provide exact, algebraic handles on Sinkhorn limits with potential implications for numerical stability, combinatorial matrix theory, and generalized scaling problems.

Abstract

We use a variety of computational tools to obtain a degree- polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive matrix. The degree of this equation has a combinatorial interpretation as the number of minors of an matrix, and the coefficients involve a determinant formula that reflects new combinatorial structure on sets of minor specifications. The tools we use include Gröbner bases, which produce equations for small matrices; the PSLQ algorithm, which produces equations for larger matrices as part of an interpolation effort that required 1.5 years of CPU time; and ChatGPT o3-mini-high, which identified the signs of the off-diagonal entries in the determinant formula.
Paper Structure (5 sections, 4 theorems, 80 equations, 2 tables)

This paper contains 5 sections, 4 theorems, 80 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Sinkhorn limit of a $3 \times 3$ matrix
  3. Square matrices and solving for coefficients
  4. Rectangular matrices and combinatorial structure
  5. Open questions

Key Result

Theorem 3

Let $A$ be a positive $3 \times 3$ matrix. The top-left entry $x$ of $\mathop{\mathrm{Sink}}\nolimits(A)$ satisfies $b_6 x^6 + \dots + b_1 x + b_0 = 0$, where the coefficients $b_k$ appear in Table 3 by 3 coefficients in factored form.

Theorems & Definitions (38)

  • Example 1
  • Conjecture 2
  • Theorem 3
  • Example 4
  • Example 5
  • Theorem 6
  • Corollary 7
  • proof
  • Corollary 8
  • proof : Proof of Corollary \ref{['3 by 3 matrix with row multiples']}
  • ...and 28 more