The Bethe Permanent of a Nonnegative Matrix

Abstract

It has recently been observed that the permanent of a nonnegative square matrix, i.e., of a square matrix containing only nonnegative 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 pseudocodewords and permanent-based kernels.

Design of improved quasi-cyclic protograph-based Raptor-like LDPC codes for short block-lengths

“…where σ refers to a permutation and the summation is over all permutations of [ ]. The permanent, although it looks deceptively similar to the determinant, is harder to compute than the determinant [9]. While the arithmetic complexity of computing the determinant is O 3 , the most efficient algorithm known to compute the permanent of any square matrix, due to Ryser [10], is of complexity Θ · 2 .…”

Belief Propagation, Bethe Approximation and Polynomials

“…This implies that if σ ∈ {0, 1} n×n corresponds to a perfect matching M in the complete graph K n,n then g(σ) = i,j A σ i,j i,j and in fact σ g(σ) = Per(A). One can show (see [36]) that the Bethe approximation for this factor graphs takes the form…”

“…where ∨ and ∧ denote entry-wise OR and entry-wise AND respectively. The proof is based on the following combinatorial characterization of the Bethe approximation, due to [36,35]. It says that…”

“…Consider the following factor graph representation of permanents [40,36]. The graph G has nodes a 1 , a 2 , .…”

A Randomized Algorithm for the Capacity of Finite-State Channels

“…This interesting algorithm has attracted a great deal of attention due to the observations that it fairly precisely approximates the channel capacity for a number of practical channels. (Notably, some results that were derived in the context of the GBAA have proven to be useful for analyzing the Bethe entropy function of some graphical models that appear in the context of low-density parity-check codes [58] and for approximately computing the permanent of a non-negative matrix [59].) For a finite-state channel, let X denote the input Markov process and Y its corresponding output process, which, by definition, is a hidden Markov process [15].…”