A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes
A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes
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关于复杂希尔伯特度量及其应用于隐马尔可夫过程熵率解析域的注释
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Y. Peres
中科院分区：
文献类型：
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作者：
G. Han;B. Marcus;Y. Peres
本文证明了正矩阵的复小扰动是标准复单纯形上关于复希尔伯特度量的压缩。我们表明，这个度量可以用来获得估计的域的分析熵率的隐马尔可夫过程时，底层马尔可夫链具有严格的正转移概率。本说明有两个目的。首先，在第1节中，我们介绍了标准真实的单纯形上的希尔伯特度量的复版本。该度量定义在标准复单纯形内的标准真实的单纯形内部的复邻域上。我们表明，如果附近是足够小的，那么对于任何足够小的复杂扰动的严格正的方阵作为一个收缩，相对于这个度量。当这篇论文接近完成时，我们被告知最近引入了一个不同的复希尔伯特度量。我们简要地讨论了这个度量[2]和我们在注1.6中的度量之间的关系。其次，我们展示了如何使用一个复杂的希尔伯特度量，以获得较低的估计域的分析熵率的隐马尔可夫过程时，底层马尔可夫链具有严格的正转移概率。解析域是重要的，因为它指定了泰勒级数收敛到熵率的显式区域，并且还给出了泰勒近似收敛速度的显式估计。
In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities. The purpose of this note is twofold. First, in Section 1, we introduce a complex version of the Hilbert metric on the standard real simplex. This metric is defined on a complex neighbourhood of the interior of the standard real simplex, within the standard complex simplex. We show that if the neighbourhood is sufficiently small, then for any sufficiently small complex perturbation of a strictly positive square matrix acts as a contraction, with respect to this metric. While this paper was nearing completion, we were informed of a different complex Hilbert metric, which was recently introduced. We briefly discuss the relation between this metric [2] and our metric in Remark 1.6. Secondly, we show how one can use a complex Hilbert metric to obtain lower estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities. The domain of analyticity is important because it specifies an explicit region where a Taylor series converges to the entropy rate and also gives an explicit estimate on the rate of convergence of the Taylor approximation.