Approximating the Permanent with Belief Propagation

This work describes a method of approximating matrix permanents efficiently using belief propagation. We formulate a probability distribution whose partition function is exactly the permanent, then use Bethe free energy to approximate this partition function. After deriving some speedups to standard belief propagation, the resulting algorithm requires O(n) time per iteration and seems empirically to take a constant number of iterations. The permanent is a scalar quantity computed from a matrix and has been an active topic of research for well over a century. It plays a role in cryptography and statistical physics where it is fundamental to Ising and dimer models. While the determinant of an n × n matrix can be evaluated exactly in sub-cubic time, efficient methods for computing the permanent have remained elusive. Since the permanent is #P-complete, efficient exact evaluations cannot be found in general. Recently, promising fully-polynomial randomized approximate schemes (FPRAS) have emerged which provide arbitrarily close approximations. Significant progress has produced an FPRAS that can handle arbitrary n× n matrices with non-negative entries [5]. The method uses Markov chain Monte Carlo and only requires a polynomial order of samples. It remains to be seen if other approximate inference methods can be brought to bear on the permanent. For instance, loopy belief propagation has also recently gained prominence in the machine learning community. The method is exact for singly-connected networks such as trees. In certain special loopy graph cases, including graphs with a single loop, bipartite matching graphs [1] and bipartite multi-matching graphs [4], the convergence of BP has been proven. In more general loopy graphs, loopy BP still maintains some surprising empirical success. Theoretical understanding of the convergence of loopy BP has recently been improved by noting certain general conditions for its fixed points and relating them to minima of Bethe free energy. This article proposes belief propagation for computing the permanent and investigates some theoretical and experimental properties. In particular, we provide evidence that BP produces a bound on the permanent for any non-negative matrix while most other known bounds require assumptions on the structure of the input matrix. We construct a bipartite factor graph similar to [1] that represents a distribution over permutations parameterized by a weight matrix. Observing that the partition function of this distribution is the permanent of the weight matrix, we approximate the permanent using Bethe free energy,