Degree-M Bethe and Sinkhorn Permanent Based Bounds on the Permanent of a Non-Negative Matrix

The permanent of a non-negative square matrix can be well approximated by finding the minimum of the Bethe free energy function associated with some suitably defined factor graph; the resulting approximation to the permanent is called the Bethe permanent. Vontobel gave a combinatorial characterization of the Bethe permanent via degree- M Bethe permanents, which is based on degree- M covers of the underlying factor graph. In this paper, we prove a degree- M -Bethe-permanent-based lower bound on the permanent of a non-negative matrix, which solves 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 → ∞, these lower and upper bounds yield known Bethe-permanent-based lower and upper bounds on the permanent of a non-negative matrix. Moreover, we prove similar results for an approximation to the permanent known as the (scaled) Sinkhorn permanent.