Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars
| | Speaker: | 郭来刚，北师大数学科学学院
| | Inviter: | | Title:
| Lower Bound on Derivatives of Costa's Differential Entropy | Time & Venue:
| 2021.12.01 14:30-15:30 腾讯会议号：169-744-781 | Abstract:
| Several conjectures concern the lower bound of the derivatives for Costa's differential entropy $H(X_t)$ of an $n$-dimensional random vector $X_t$ depending on time $t$. Cheng and Geng conjectured that $H(X_t)$ is completely monotone in $t$, that is, $C_1(m,n): (-1)^{m+1}(\d^m/\d^m t)H(X_t)\ge0$. 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, $C_2(m,n): (-1)^{m+1}(\d^m/\d^m t)H(X_t)\ge(-1)^{m+1}(\d^m/\d^m t)H(X_{Gt})$. For $C_1(m,n)$, Costa proved $C_1(1,n)$ and $C_1(2,n)$ and Cheng-Geng proved $C_1(3,1)$ and $C_1(4,1)$. 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. Notice that $C_2(2,n)$ generalizes McKean's result from univariate case to multivariate case, also generalizes Costa's result $C_1(2,n)$ in that the precise lower bound for the second order derivative of $H(X_t)$ is given. We further observe that McKean's conjecture might not be true for $n>1$ and $m>2$ and propose a weaker version $C_3(m,n): (-1)^{m+1}(\d^m/\d^mt)H(X_t)\ge(-1)^{m+1}\frac{1}{n}(\d^m/\d^m t)H(X_{Gt})$. We prove $C_3(3,2)$, $C_3(3,3)$, $C_3(3,4)$, $C_3(4,2)$ under the log-concave condition.
In this talk, we prove new results about these three conjectures. A systematic procedure to prove $C_l(m,n)$ is proposed based on semidefinite programming and the results mentioned above are proved using this procedure.
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