Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy
文献类型：期刊论文
| 作者 | Guo, Laigang1; Yuan, Chun-Ming2,3; Gao, Xiao-Shan2,3 |
| 刊名 | ENTROPY
 |
| 出版日期 | 2022-08-01 |
| 卷号 | 24期号:8页码:17 |
| 关键词 | differential entropy completely monotone Mckean's conjecture log-concavity Gaussian optimality |
| DOI | 10.3390/e24081155 |
| 英文摘要 | This paper studies the properties of the derivatives of differential entropy H(X-t) in Costa's entropy power inequality. For real-valued random variables, Cheng and Geng conjectured that for m >= 1, (-1)(m+1)(d(m)/dt(m))H(X-t) >= 0, while McKean conjectured a stronger statement, whereby (-1)(m+1)(d(m)/dt(m))H(X-t) >= (-1)(m+1)(d(m)/dt(m))H(X-Gt). Here, we study the higher dimensional analogues of these conjectures. In particular, we study the veracity of the following two statements: C-1(m, n) : (-1)(m+1)(d(m)/dt(m))H(X-t) >= 0, where n denotes that X-t is a random vector taking values in R-n, and similarly, C-2(m, n) : (-1)(m+1)(d(m)/dt(m))H(X-t) >= (-1)(m+1)(d(m)/dt(m))H(X-Gt) >= 0. In this paper, we prove some new multivariate cases: C-1(3,i), i = 2,3,4. Motivated by our results, we further propose a weaker version of McKean's conjecture C-3(m, n) : (-1)(m+1)(d(m)/dt(m))H(X-t) >= (-1)(m+1)1/n(d(m)/dt(m))H(X-Gt), which is implied by C-2(m,n) and implies C-1(m,n). We prove some multivariate cases of this conjecture under the log-concave condition: C-3(3,i),i = 2,3,4 and C-3(4,2). A systematic procedure to prove C-l(m,n) is proposed based on symbolic computation and semidefinite programming, and all the new results mentioned above are explicitly and strictly proved using this procedure. |
| 资助项目 | NSFC[11688101] ; NSFC[NKRDP 2018YFA0704705] ; Beijing Natural Science Foundation[Z190004] ; Fundamental Research Funds for the Central Universities[2021NTST32] |
| WOS研究方向 | Physics |
| 语种 | 英语 |
| 出版者 | MDPI |
| WOS记录号 | WOS:000846004200001 |
| 源URL | [http://ir.amss.ac.cn/handle/2S8OKBNM/61059]  |
| 专题 | 中国科学院数学与系统科学研究院 |
| 通讯作者 | Gao, Xiao-Shan |
| 作者单位 | 1.Beijing Normal Univ, Lab Math & Complex Syst, Minist Educ, Sch Math Sci, Beijing 100875, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, KLMM, Beijing 100190, Peoples R China 3.Univ Chinese Acad Sci, Beijing 100049, Peoples R China |
| 推荐引用方式 GB/T 7714 | Guo, Laigang,Yuan, Chun-Ming,Gao, Xiao-Shan. Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy[J]. ENTROPY,2022,24(8):17. |
| APA | Guo, Laigang,Yuan, Chun-Ming,&Gao, Xiao-Shan.(2022).Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy.ENTROPY,24(8),17. |
| MLA | Guo, Laigang,et al."Lower Bounds on Multivariate Higher Order Derivatives of Differential Entropy".ENTROPY 24.8(2022):17. |
入库方式： OAI收割
来源：数学与系统科学研究院
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