论文标题
哥斯达黎加零熵的衍生物的下限
Lower Bound on Derivatives of Costa's Differential Entropy
论文作者
论文摘要
几个猜想涉及$ n $二维随机矢量的差分熵$ h（x_t）$的下限。 Cheng和Geng猜想$ 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）h（x_t）\ ge0 $。 McKean猜想Gaussian $ x_ {gt} $达到了$（-1）^{m+1}（d^m/d^m t）h（x_t）$的最低t）h（x_t）\ ge（-1）^{m+1}（d^m/d^m t）h（x_ {gt}）$。麦基恩的猜想仅在单变量案例中考虑：$ C_2（1,1）$和$ C_2（2,1）$由McKean和$ C_2（I，1）证明，i = 3,4,5 $在登录concave条件下证明了Zhang-anantharam-geng的证明。在本文中，我们证明了$ C_2（1，N）$，$ C_2（2，N）$，并观察到McKean的猜想可能不是$ n> 1 $和$ m> 2 $的猜想。我们进一步提出了一个较弱的版本$ C_3（m，n）：（-1）^{m+1}（d^m/d^m t） $ c_3（3,3）$，$ c_3（3,4）$，$ c_3（4,2）$在log-concave条件下。基于半决赛编程提出了证明$ C_L（M，N）$的系统过程，并使用此过程证明了上述结果。
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.
