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Degree-$M$ Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix

Yuwen Huang, Navin Kashyap, Pascal O. Vontobel

TL;DR

This work establishes degree-M Bethe- and degree-M scaled Sinkhorn-permanent-based lower and upper bounds on the permanent of a non-negative matrix, resolving a conjecture of Vontobel. By representing the permanent via an S-NFG and analyzing its M-cover lifts, the authors derive coefficient recursions C_{M,n}(γ), C_{B,M,n}(γ), and C_{scS,M,n}(γ) that express (perm(Θ))^M, (perm_B,M(Θ))^M, and (perm_scS,M(Θ))^M as sums over Γ_{M,n}, and prove explicit inequalities bounding the ratios of these coefficients. They further show asymptotic characterizations of the coefficients through entropy-like functions H'_G, H_B, and H_scS, and demonstrate that the degree-M quantities converge to the classic Bethe and Sinkhorn bounds as M→∞. The paper also provides a concrete M=2 result linking perm/perm_B,2 to pairwise permutation statistics and outlines open problems for elementary proofs, bound tightening, and generalization to broader S-NFG classes. Overall, the work deepens the combinatorial understanding of Bethe and Sinkhorn approximations for the matrix permanent and offers a rigorous path from finite-M covers to known limiting bounds.

Abstract

The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree-$M$ Bethe permanents, which are based on degree-$M$ covers of the underlying factor graph. In this paper, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree-$M$-Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit $M \to \infty$, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.

Degree-$M$ Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative Matrix

TL;DR

This work establishes degree-M Bethe- and degree-M scaled Sinkhorn-permanent-based lower and upper bounds on the permanent of a non-negative matrix, resolving a conjecture of Vontobel. By representing the permanent via an S-NFG and analyzing its M-cover lifts, the authors derive coefficient recursions C_{M,n}(γ), C_{B,M,n}(γ), and C_{scS,M,n}(γ) that express (perm(Θ))^M, (perm_B,M(Θ))^M, and (perm_scS,M(Θ))^M as sums over Γ_{M,n}, and prove explicit inequalities bounding the ratios of these coefficients. They further show asymptotic characterizations of the coefficients through entropy-like functions H'_G, H_B, and H_scS, and demonstrate that the degree-M quantities converge to the classic Bethe and Sinkhorn bounds as M→∞. The paper also provides a concrete M=2 result linking perm/perm_B,2 to pairwise permutation statistics and outlines open problems for elementary proofs, bound tightening, and generalization to broader S-NFG classes. Overall, the work deepens the combinatorial understanding of Bethe and Sinkhorn approximations for the matrix permanent and offers a rigorous path from finite-M covers to known limiting bounds.

Abstract

The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy functions associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree- Bethe permanents, which are based on degree- covers of the underlying factor graph. In this paper, we prove a degree--Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves a conjecture proposed by Vontobel in [IEEE Trans. Inf. Theory, Mar. 2013]. We also prove a degree--Bethe-permanent-based upper bound on the permanent of a non-negative matrix. In the limit , these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.
Paper Structure (17 sections, 21 theorems, 92 equations, 6 figures)

This paper contains 17 sections, 21 theorems, 92 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Basic Definitions and Notations
  4. An S-NFG Representation of the Permanent
  5. Free Energy Functions Associated with
  6. Recursions for
  7. Bounding the Permanent of
  8. Asymptotic Expressions for the Coefficients
  9. Results for the Permanent and the
  10. Conclusion and Open Problems
  11. Proof of Proposition \ref{['sec:1:prop:14']}
  12. Proof of Lemma \ref{['sec:1:lem:15:copy:2']}
  13. Proof of Lemma \ref{['sec:1:lem:30']}
  14. Proof of Lemma \ref{['lemma:ratio:perm:permbM:1']}
  15. An Alternative proof of the inequalities in \ref{['sec:1:eqn:147']} in Theorem \ref{['th:main:permanent:inequalities:1']}
  16. ...and 2 more sections

Key Result

Lemma 2

There are collections of non-negative real numbers $\bigl\{ C_{M,n}( \bm{\gamma} ) \bigr\}_{ \bm{\gamma} \in \Gamma_{M,n} }$, $\bigl\{ C_{\mathrm{B},M,n}( \bm{\gamma} ) \bigr\}_{ \bm{\gamma} \in \Gamma_{M,n} }$, $\bigl\{ C_{\mathrm{scS},M,n}( \bm{\gamma} ) \bigr\}_{ \bm{\gamma $\square$

Figures (6)

  • Figure 1: Combinatorial characterizations of the Bethe and scaled Sinkhorn permanents.
  • Figure 2: The S-NFG $\mathsf{N} ( \bm{\theta} )$ for the special case $n = 3$.
  • Figure 3: Generalization of Pascal's triangle visualizing the recursion in Lemma \ref{['sec:1:lem:29:copy:2']}.
  • Figure 4: Generalization of Pascal's triangle visualizing the recursion in Lemma \ref{['sec:1:lem:15:copy:2']}.
  • Figure 5: Generalization of Pascal's triangle visualizing the recursion in Lemma \ref{['sec:1:lem:30']}.
  • ...and 1 more figures

Theorems & Definitions (57)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Definition 6
  • Definition 7
  • Lemma 8
  • proof
  • Definition 9
  • ...and 47 more