论文标题
证明了Costa的熵功率不平等和具有半决赛差异熵的高阶不平等
Prove Costa's Entropy Power Inequality and High Order Inequality for Differential Entropy with Semidefinite Programming
论文作者
论文摘要
哥斯达黎加的熵力不平等是对香农熵力不平等的重要概括。与Costa的熵功能不平等和McKean在1966年提出的猜想有关，Cheng-Geng最近推测$ d（m，n）：（-1）^{m+1}（\ partial^m/\ partial^m t） $ h（x_t）$ $ x_t $的微分熵。 $ d（1，n）$和$ d（2，n）$被科斯塔证明是科斯塔熵不平等的后果。 Cheng-Geng证明$ D（3,1）$和$ D（4,1）$。在本文中，我们提出了一个系统的程序，以证明$ d（m，n）$和哥斯达黎加的熵功能不平等，基于半节目。使用基于此过程的软件包，我们证明了$ d（3，n）$，$ n = 2,3,4 $，并为Costa的熵功率不平等提供了新的证明。我们还表明，使用当前已知的约束，该程序无法证明$ d（5,1）$和$ d（4,2）$。
Costa's entropy power inequality is an important generalization of Shannon's entropy power inequality. Related with Costa's entropy power inequality and a conjecture proposed by McKean in 1966, Cheng-Geng recently conjectured that $D(m,n): (-1)^{m+1}(\partial^m/\partial^m t)H(X_t)\ge0$, where $X_t$ is the $n$-dimensional random variable in Costa's entropy power inequality and $H(X_t)$ the differential entropy of $X_t$. $D(1,n)$ and $D(2,n)$ were proved by Costa as consequences of Costa's entropy power inequality. Cheng-Geng proved $D(3,1)$ and $D(4,1)$. In this paper, we propose a systematical procedure to prove $D(m,n)$ and Costa's entropy power inequality based on semidefinite programming. Using software packages based on this procedure, we prove $D(3,n)$ for $n=2,3,4$ and give a new proof for Costa's entropy power inequality. We also show that with the currently known constraints, $D(5,1)$ and $D(4,2)$ cannot be proved with the procedure.
