Approximating the Permanent with Fractional Belief Propagation

TL;DR

This paper explores fractional belief propagation methods for approximating the permanent of non-negative matrices, analyzing theoretical bounds and proposing a new parameterized functional with practical estimation guidance.

Contribution

It introduces a novel fractional belief propagation approach with a parameterized free energy functional, providing new bounds, conjectures, and empirical insights for permanent approximation.

Findings

The fractional free energy functional is monotonic and continuous in the parameter b3.

A special b3_* value is defined where the functional equals the matrix permanent.

Empirical results show b3_* varies across ensembles but remains within b3 d [-1, -1/2], guiding practical permanent estimation.

Abstract

We discuss schemes for exact and approximate computations of permanents, and compare them with each other. Specifically, we analyze the Belief Propagation (BP) approach and its Fractional Belief Propagation (FBP) generalization for computing the permanent of a non-negative matrix. Known bounds and conjectures are verified in experiments, and some new theoretical relations, bounds and conjectures are proposed. The Fractional Free Energy (FFE) functional is parameterized by a scalar parameter $\gamma\in[-1;1]$, where $\gamma=-1$ corresponds to the BP limit and $\gamma=1$ corresponds to the exclusion principle (but ignoring perfect matching constraints) Mean-Field (MF) limit. FFE shows monotonicity and continuity with respect to $\gamma$. For every non-negative matrix, we define its special value $\gamma_*\in[-1;0]$ to be the $\gamma$ for which the minimum of the $\gamma$-parameterized FFE functional is equal to the permanent of the matrix, where the lower and upper bounds of the $\gamma$-interval corresponds to respective bounds for the permanent. Our experimental analysis suggests that the distribution of $\gamma_*$ varies for different ensembles but $\gamma_*$ always lies within the $[-1;-1/2]$ interval. Moreover, for all ensembles considered the behavior of $\gamma_*$ is highly distinctive, offering an emprirical practical guidance for estimating permanents of non-negative matrices via the FFE approach.