雷尼熵幂凹性的一种推广。
A Generalization of the Concavity of Rényi Entropy Power.
作者信息
Guo Laigang, Yuan Chun-Ming, Gao Xiao-Shan
机构信息
Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China.
KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China.
出版信息
Entropy (Basel). 2021 Nov 27;23(12):1593. doi: 10.3390/e23121593.
Recently, Savaré-Toscani proved that the Rényi entropy power of general probability densities solving the -nonlinear heat equation in Rn is a concave function of time under certain conditions of three parameters n,p,μ, which extends Costa's concavity inequality for Shannon's entropy power to the Rényi entropy power. In this paper, we give a condition Φ(n,p,μ) of n,p,μ under which the concavity of the Rényi entropy power is valid. The condition Φ(n,p,μ) contains Savaré-Toscani's condition as a special case and much more cases. Precisely, the points (n,p,μ) satisfying Savaré-Toscani's condition consist of a two-dimensional subset of R3, and the points satisfying the condition Φ(n,p,μ) consist a three-dimensional subset of R3. Furthermore, Φ(n,p,μ) gives the necessary and sufficient condition in a certain sense. Finally, the conditions are obtained with a systematic approach.
最近，萨瓦雷 - 托斯卡尼证明，在三个参数(n)、(p)、(\mu)的某些条件下，求解(\mathbb{R}^n)中(p -)非线性热方程的一般概率密度的雷尼熵幂是时间的凹函数，这将科斯塔关于香农熵幂的凹性不等式推广到了雷尼熵幂。在本文中，我们给出了(n)、(p)、(\mu)的一个条件(\varPhi(n,p,\mu))，在该条件下雷尼熵幂的凹性成立。条件(\varPhi(n,p,\mu))包含萨瓦雷 - 托斯卡尼的条件作为特殊情况以及更多情况。确切地说，满足萨瓦雷 - 托斯卡尼条件的点((n,p,\mu))构成(\mathbb{R}^3)的一个二维子集，而满足条件(\varPhi(n,p,\mu))的点构成(\mathbb{R}^3)的一个三维子集。此外，在某种意义上，(\varPhi(n,p,\mu))给出了充要条件。最后，这些条件是通过一种系统的方法得到的。
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