Lower Bound on Derivatives of Costa's Differential Entropy

Abstract

Several conjectures concern the lower bound for the differential entropy $H(X_t)$ of an $n$-dimensional random vector $X_t$ introduced by Costa. Cheng and Geng conjectured that $H(X_t)$ is completely monotone, that is, $C_1(m,n):(-1{)}^{m+1}(d^m/d^{m}t)H(X_t)\ge 0$. McKean conjectured that Gaussian $X_{Gt}$ achieves the minimum of $(-1{)}^{m+1}(d^m/d^{m}t)H(X_t)$ under certain conditions, that is, $C_2(m,n):(-1{)}^{m+1}(d^m/d^{m}t)H(X_t)\ge (-1{)}^{m+1}(d^m/d^{m}t)H(X_{Gt})$. McKean's conjecture was only considered in the univariate case before: $C_2(1,1)$ and $C_2(2,1)$ were proved by McKean and $C_2(i,1),i=3,4,5$ were proved by Zhang-Anantharam-Geng under the log-concave condition. In this paper, we prove $C_2(1,n)$, $C_2(2,n)$ and observe that McKean's conjecture might not be true for $n>1$ and $m>2$. We further propose a weaker version $C_3(m,n):(-1{)}^{m+1}(d^m/d^{m}t)H(X_t)\ge (-1{)}^{m+1}\frac{1}{n}(d^m/d^{m}t)H(X_{Gt})$ and prove $C_3(3,2)$, $C_3(3,3)$, $C_3(3,4)$, $C_3(4,2)$ under the log-concave condition. A systematical procedure to prove $C_l(m,n)$ is proposed based on semidefinite programming and the results mentioned above are proved using this procedure.

arXiv admin note: text overlap with arXiv:2004.08543

References in corpus (2)

• A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI
• Prove Costa's Entropy Power Inequality and High Order Inequality for Differential Entropy with Semidefinite Programming