Bounding the Bethe and the Degree-$M$ Bethe Permanents

It was recently conjectured that the permanent of a ${P}$-lifting $θ^{\uparrow{P}}$ of a matrix $θ$ of degree $M$ is less than or equal to the $M$th power of the permanent perm$(θ)$, i.e., perm$(θ^{\uparrow{P}})\leq(\text{perm}(θ))^M$ and, consequently, that the degree-$M$ Bethe permanent $\text{perm}_{M,\mathrm{B}} (θ)$ of a matrix $θ$ is less than or equal to the permanent perm$(θ)$ of $θ$, i.e., perm$_{M, \mathrm{B}} (θ)\leq \text{perm}(θ)$. In this paper, we prove these related conjectures and show in addition a few 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$_{\mathrm{B}} (θ)\leq \text{perm}(θ)$ on the Bethe permanent of the base matrix $θ$ that uses only the combinatorial definition of the Bethe permanent.

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