Bethe and M-Bethe Permanent Inequalities

It was recently conjectured that the permanent of a $P$-lifting $\theta^{\uparrow P}$ of a matrix $\theta$ of degree $M$ is less than or equal to the $M$th power of the permanent perm$(\theta)$, i.e., perm$(\theta^{\uparrow P})\leq$perm$(\theta)^M$ and, consequently, that the degree-$M$ Bethe permanent perm$_{M,\mathrm{B}} (\theta)$ of a matrix $\theta$ is less than or equal to the permanent perm$(\theta)$ of $\theta$, i.e., perm$_{M, \mathrm{B}} (\theta)\leq$perm$(\theta)$. In this paper, we prove these related conjectures and show some properties of the permanent of block matrices that are lifts of a matrix. As a corollary, we obtain an alternative proof of the inequality perm$_{\mathrm{B}} (\theta)\leq$perm$(\theta)$ on the Bethe permanent of the base matrix $\theta$, which uses only the combinatorial definition of the Bethe-permanent. The results have implications in coding theory. Since a $P$-lifting corresponds to an $M$-graph cover and thus to a protograph-based LDPC code, the results may help explain the performance of these codes.