Weighted Gaussian entropy and determinant inequalities
Weighted Gaussian entropy and determinant inequalities
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DOI：
10.1007/s00010-021-00861-3
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发表时间：
2022-01
影响因子：
0.8
通讯作者：
I. Stuhl;M. Kelbert;Y. Suhov;S. Yasaei Sekeh
中科院分区：
文献类型：
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作者：
I. Stuhl;M. Kelbert;Y. Suhov;S. Yasaei Sekeh
我们将信息论的不平等（Dembo-Cover-Thomas在1988-1991年讨论过）扩展到熵的加权版本，得出了一系列结果。大多数产生的不等式涉及高斯加权熵；它们暗示了正定矩阵行列式的一些新关系。与香农熵不同，香农熵中结果的贡献仅取决于其概率，加权熵（或上下文相关）熵考虑了由给定权重函数决定的结果的“值”。一个新结果的例子是正定矩阵的行列式与其不同大小的方块（子矩阵）之间的强Hadamard不等式（SHI）的加权版本。当，加权不等式成为“标准”SHI；一般来说，加权版本需要一些假设。SHI及其加权版本概括了一个广为人知的“通常”Hadamard不等式。
We produce a series of results extending information-theoretical inequalities (discussed by Dembo–Cover–Thomas in 1988–1991) to a weighted version of entropy. Most of the resulting inequalities involve the Gaussian weighted entropy; they imply a number of new relations for determinants of positive-definite matrices. Unlike the Shannon entropy where the contribution of an outcome depends only upon its probability, the weighted (or context-dependent) entropy takes into account a ‘value’ of an outcome determined by a given weight function. An example of a new result is a weighted version of the strong Hadamard inequality (SHI) between the determinants of a positive-definitematrix and its square blocks (sub-matrices) of different sizes. When, the weighted inequality becomes a ‘standard’ SHI; in general, the weighted version requires some assumptions upon. The SHI and its weighted version generalize a widely known ‘usual’ Hadamard inequality.