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Finite-Graph-Cover-Based Analysis of Factor Graphs in Classical and Quantum Information Processing Systems

Yuwen Huang

TL;DR

This work develops a unified framework using finite graph covers to analyze the sum-product algorithm and Bethe-type approximations for both standard (classical) factor graphs and double-edge (quantum) factor graphs. It introduces degree-$M$ Bethe permanents and degree-$M$ Sinkhorn permanents as graph-cover-based surrogates for the exact matrix permanent, proving lower and upper bounds that converge to known Bethe/Sinkhorn limits as $M\to\infty$. For DE-FGs, it provides a combinatorial characterization of the Bethe partition function under an easily checkable condition by employing a novel loop-calculus transform, and it extends this with the symmetric-subspace transform to express these characterizations as integrals, with SST also applicable to DE-FGs. The combination of LCT and SST broadens the analytical toolbox for both S-FGs and DE-FGs and connects Bethe approximations to graph-cover statistics, with implications for classical inference and quantum information processing via tensor-network frameworks.

Abstract

In this thesis, we leverage finite graph covers to analyze the SPA and the Bethe partition function for both S-FGs and DE-FGs. There are two main contributions in this thesis. The first main contribution concerns a special class of S-FGs where the partition function of each S-FG equals the permanent of a nonnegative square matrix. The Bethe partition function for such an S-FG is called the Bethe permanent. A combinatorial characterization of the Bethe permanent is given by the degree-$M$ Bethe permanent, which is defined based on the degree-$M$ graph covers of the underlying S-FG. In this thesis, we prove a degree-$M$-Bethe-permanent-based lower bound on the permanent of a non-negative square matrix, resolving 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 square matrix. The second main contribution is giving a combinatorial characterization of the Bethe partition function for DE-FGs in terms of finite graph covers. In general, approximating the partition function of a DE-FG is more challenging than for an S-FG because the partition function of the DE-FG is a sum of complex values and not just a sum of non-negative real values. Moreover, one cannot apply the method of types for proving the combinatorial characterization as in the case of S-FGs. We overcome this challenge by applying a suitable loop-calculus transform (LCT) for both S-FGs and DE-FGs. Currently, we provide a combinatorial characterization of the Bethe partition function in terms of finite graph covers for a class of DE-FGs satisfying an (easily checkable) condition.

Finite-Graph-Cover-Based Analysis of Factor Graphs in Classical and Quantum Information Processing Systems

TL;DR

This work develops a unified framework using finite graph covers to analyze the sum-product algorithm and Bethe-type approximations for both standard (classical) factor graphs and double-edge (quantum) factor graphs. It introduces degree- Bethe permanents and degree- Sinkhorn permanents as graph-cover-based surrogates for the exact matrix permanent, proving lower and upper bounds that converge to known Bethe/Sinkhorn limits as . For DE-FGs, it provides a combinatorial characterization of the Bethe partition function under an easily checkable condition by employing a novel loop-calculus transform, and it extends this with the symmetric-subspace transform to express these characterizations as integrals, with SST also applicable to DE-FGs. The combination of LCT and SST broadens the analytical toolbox for both S-FGs and DE-FGs and connects Bethe approximations to graph-cover statistics, with implications for classical inference and quantum information processing via tensor-network frameworks.

Abstract

In this thesis, we leverage finite graph covers to analyze the SPA and the Bethe partition function for both S-FGs and DE-FGs. There are two main contributions in this thesis. The first main contribution concerns a special class of S-FGs where the partition function of each S-FG equals the permanent of a nonnegative square matrix. The Bethe partition function for such an S-FG is called the Bethe permanent. A combinatorial characterization of the Bethe permanent is given by the degree- Bethe permanent, which is defined based on the degree- graph covers of the underlying S-FG. In this thesis, we prove a degree--Bethe-permanent-based lower bound on the permanent of a non-negative square matrix, resolving 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 square matrix. The second main contribution is giving a combinatorial characterization of the Bethe partition function for DE-FGs in terms of finite graph covers. In general, approximating the partition function of a DE-FG is more challenging than for an S-FG because the partition function of the DE-FG is a sum of complex values and not just a sum of non-negative real values. Moreover, one cannot apply the method of types for proving the combinatorial characterization as in the case of S-FGs. We overcome this challenge by applying a suitable loop-calculus transform (LCT) for both S-FGs and DE-FGs. Currently, we provide a combinatorial characterization of the Bethe partition function in terms of finite graph covers for a class of DE-FGs satisfying an (easily checkable) condition.

Paper Structure

This paper contains 65 sections, 53 theorems, 235 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-negative square Matrix
  3. The Permanent of a Square Matrix
  4. The Bethe Permanent and the (Scaled) Sinkhorn Permanent of a Non-negative Square Matrix
  5. A Combinatorial Characterization of the Bethe Permanent and the (Scaled) Sinkhorn Permanent
  6. Main Contributions
  7. Characterizing the Bethe Partition Function of Double-Edges Factor Graphs via Graph Covers
  8. Main Contributions
  9. Structure of this Thesis
  10. Preliminaries
  11. Basic Definition and Notations
  12. Standard Normal Factor Graphs (S-NFGs)
  13. The S-NFG for the Permanent of a Non-negative Square Matrix
  14. Free Energy Functions Associated with
  15. Double-Edge Normal Factor Graphs (DE-NFGs)
  16. ...and 50 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 (16)

  • Figure 1: A combinatorial characterization of the Bethe partition function for an S-FG.
  • Figure 2: Combinatorial characterizations of the Bethe and scaled Sinkhorn permanents.
  • Figure 3: NFG $\mathsf{N}$ in Example \ref{['sec:SNFG:exp:1']}.
  • Figure 5: The S-NFG $\mathsf{N} ( \bm{\theta} )$ for the special case $n = 3$.
  • Figure 6: DE-NFG in Example \ref{['sec:DENFG:exp:1']}.
  • ...and 11 more figures

Theorems & Definitions (158)

  • Definition 1
  • Lemma 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Example 6
  • Example 7
  • Definition 8
  • Definition 9
  • Proposition 10
  • ...and 148 more