A proof of the log-concavity conjecture related to the computation of the ergodic capacity of MIMO channels

TL;DR

This paper proves a conjecture about the log-concavity of a polynomial related to the ergodic capacity of MIMO channels, enabling efficient optimization under certain conditions.

Contribution

It provides a self-contained proof of the log-concavity conjecture using H-Stable polynomial theory, facilitating capacity computation.

Findings

Proof of the log-concavity conjecture for the polynomial p

Enabling efficient maximization of the ergodic capacity bound

Application of H-Stable polynomial theory to MIMO channel analysis

Abstract

An upper bound on the ergodic capacity of {\bf MIMO} channels was introduced recently in arXiv:0903.1952. This upper bound amounts to the maximization on the simplex of some multilinear polynomial $p(\lambda_1,...,\lambda_n)$ with non-negative coefficients. Interestingly, the coefficients are subpermanents of some non-negative matrix. In general, such maximizations problems are {\bf NP-HARD}. But if say, the functional $\log(p)$ is concave on the simplex and can be efficiently evaluated, then the maximization can also be done efficiently. Such log-concavity was conjectured in arXiv:0903.1952. We give in this paper self-contained proof of the conjecture, based on the theory of {\bf H-Stable} polynomials.