Logarithmic Sobolev inequalities and strong data processing theorems for discrete channels
Logarithmic Sobolev inequalities and strong data processing theorems for discrete channels
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离散通道的对数 Sobolev 不等式和强数据处理定理
DOI：
10.1109/isit.2013.6620260
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
M. Raginsky
中科院分区：
文献类型：
--
作者：
M. Raginsky
通过比较输入和输出分布的适当泛函，可以测量通道的噪声。例如，如果我们固定了一个参考输入分布，那么对于任何其他输入分布，输出相对熵与输入相对熵的最坏情况比由数据处理定理限制为1。然而，对于固定的参考输入分布，这个量可能严格小于1，给出所谓的强数据处理不等式（SDPI）。本文表明，在SDPI和实现它的任何输入分布确定最佳常数的问题，可以解决使用所谓的对数Sobolev不等式，其中涉及输入相对熵的某些措施的输入-输出相关性。另一个贡献是证明了SDPI和某些强数据处理不等式的限制情况下的雷诺分歧之间的等价性。
The noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions. For instance, if we fix a reference input distribution, then the worst-case ratio of output relative entropy to input relative entropy for any other input distribution is bounded by one, by the data processing theorem. However, for a fixed reference input distribution, this quantity may be strictly smaller than one, giving so-called strong data processing inequalities (SDPIs). This paper shows that the problem of determining both the best constant in an SDPI and any input distributions that achieve it can be addressed using so-called logarithmic Sobolev inequalities, which relate input relative entropy to certain measures of input-output correlation. Another contribution is a proof of equivalence between SDPIs and a limiting case of certain strong data processing inequalities for the Rényi divergence.