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arXiv 0911.0696cs.ITmath.IT
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与MIMO信道遍历容量计算有关的log-drop猜想的证明

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

Leonid Gurvits

摘要:

最近在arXiv:0903.1952中引入了关于{ \bf MIMO}信道的遍历容量的上限。这个上界等于某个非负系数的多线性多项式$p(λ_1,...,λ_n)$的单形的最大化。有趣的是,系数是某个非负矩阵的次积式。通常,这样的最大化问题是{ \bf NP-HARD}。但是如果说,泛函$\log(p)$在单纯形上是凹的,并且可以被有效地求值,那么最大化也可以被有效地完成。在arXiv:0903.1952中记录了这种对数。本文基于{ \bf H-稳定}多项式理论,给出了该猜想的完备证明.

英文摘要:

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 $p(λ_1,...,λ_n)$ 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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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