Prove Costa's Entropy Power Inequality and High Order Inequality for Differential Entropy with Semidefinite Programming

04/18/2020 ∙ by Laigang Guo, et al. ∙ 0

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(∂^m/∂^m t)H(X_t)>0, 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.

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