Costa微分熵导数的下界
Lower Bound on Derivatives of Costa's Differential Entropy
摘要：
Costa提出了$n$维随机向量$X_t$的微分熵$H(X_t)$的下界. Cheng和Geng证明了$H(X_t)$是完全单调的，即$C_1(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge0$。McKean证明了高斯$X_{Gt}$在一定条件下达到了$(-1)^{m+1}(d^m/d^m t)H(X_t)$的最小值，即$C_2(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge(-1)^{m+1}(d^m/d^m t)H(X_{Gt})$。McKean猜想以前只考虑了单变量情形：在对数凹条件下，McKean证明了$C_2(1,1)$和$C_2(2,1)$，Zhang-Anantharam-Geng证明了$C_2(i,1),i=3,4,5$.本文证明了$C_2(1,n)$，$C_2(2,n)$，并指出McKean猜想对$n>1$和$m>2$可能不成立.我们进一步提出了一个较弱的版本$C_3(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge(-1)^{m+1}\frac{1}{n}(d^m/d^m t)H(X_{Gt})$，并在对数凹条件下证明了$C_3(3,2)$，$C_3(3,3)$，$C_3(3,4)$，$C_3(4,2)$.基于半定规划理论，给出了$C_l(m,n)$的一个系统的证明过程，并利用该过程对上述结果进行了证明。
英文摘要：
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, $C_1(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge0$. 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, $C_2(m,n): (-1)^{m+1}(d^m/d^m t)H(X_t)\ge(-1)^{m+1}(d^m/d^m t)H(X_{Gt})$. 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.
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