The Bethe Permanent of a Non-Negative Matrix

TL;DR

This paper explores the Bethe permanent as an efficient approximation method for the permanent of non-negative matrices, demonstrating its convexity, computational efficiency, and potential for improved bounds and applications.

Contribution

It proves the convexity of the Bethe free energy, shows the sum-product algorithm efficiently finds its minimum, and provides a combinatorial characterization of the Bethe permanent.

Findings

Bethe free energy function is convex.

Sum-product algorithm efficiently finds the minimum.

Bethe permanent lower bounds the permanent.

Abstract

It has recently been observed that the permanent of a non-negative square matrix, i.e., of a square matrix containing only non-negative real entries, can very well be approximated by solving a certain Bethe free energy function minimization problem with the help of the sum-product algorithm. We call the resulting approximation of the permanent the Bethe permanent. In this paper we give reasons why this approach to approximating the permanent works well. Namely, we show that the Bethe free energy function is convex and that the sum-product algorithm finds its minimum efficiently. We then discuss the fact that the permanent is lower bounded by the Bethe permanent, and we comment on potential upper bounds on the permanent based on the Bethe permanent. We also present a combinatorial characterization of the Bethe permanent in terms of permanents of so-called lifted versions of the matrix under consideration. Moreover, we comment on possibilities to modify the Bethe permanent so that it approximates the permanent even better, and we conclude the paper with some observations and conjectures about permanent-based pseudo-codewords and permanent-based kernels.