Ergodicity Coefficients are Induced Matrix Seminorms
Ergodicity Coefficients are Induced Matrix Seminorms
复制标题
遍历系数是归纳矩阵半范数
DOI：
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发表时间：
2022
期刊：
影响因子：
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通讯作者：
M. E. Valcher
中科院分区：
文献类型：
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作者：
G. Pasquale;F. Bullo;M. E. Valcher
遍历系数是研究非齐次马氏链收敛性和平均算法的一个有用的代数工具。他们的研究历史悠久，可以追溯到马尔科夫的原著。在这项工作中，我们证明了遍历系数是如何等于一定的诱导矩阵范数和诱导范数的最优收缩矩阵。这种等价性澄清了它们作为压缩动力系统的收缩因子的用途。特别地，证明了Dobrushin遍历系数τ1等于两个不同的诱导矩阵范数.在马尔可夫链的背景下，我们提供了一个表达式的混合时间的l∞遍历系数。最后，我们表明，对于原始矩阵，诱导矩阵范数最小化的效率差距估计的收敛因子的牵引动力系统，从而证明是一个更准确的工具遍历系数相比。
Ergodicity coefficients are a useful algebraic tool in the study of the convergence properties of inhomogeneous Markov chains and averaging algorithms. Their study has a rich history, going back all the way to the original works by Markov. In this work we show how ergodicity coefficients are equal to certain induced matrix seminorms and the induced norm of optimally-deflated matrices. This equivalence clarifies their use as contraction factors for semicontractive dynamical systems. In particular, the Dobrushin ergodicity coefficient τ1 is shown to be equal to two different induced matrix seminorms. In the context of Markov chains, we provide an expression for the mixing time in terms of l∞ ergodicity coefficient. Finally, we show that, for primitive matrices, induced matrix seminorms minimize the efficiency gap in the estimation of the convergence factor of semicontractive dynamical systems, thus proving to be a more accurate tool compared to ergodicity coefficients.