arXiv 2111.062012021-11-11math.PR

块马尔可夫链随机矩阵的谱范数界

Spectral norm bounds for block Markov chain random matrices

Jaron Sanders, Albert Senen-Cerda

备注：: 30 pages, 1 figure

摘要：

本文对由块马尔可夫链（BMC）的路径构成的中心随机矩阵的最大奇异值的渐近阶进行了量化。在BMC中，存在$n$标记的状态，每个状态与$K$簇中的一个相关联，并且跳转的概率仅取决于起源和目的地的簇。给定一条从平衡态出发的路径$X_0, X_1, \ldots, X_{T_n}$，我们构造一个随机矩阵$\hat{N}$，它记录了每对状态之间的跃迁次数。证明了若$ω(n) = T_n = o(n^2)$，则$\| \hat{N} - \mathbb{E}[\hat{N}] \| = Ω_{\mathbb{P}}(\sqrt{T_n/n})$ .我们还证明，如果$T_n = Ω(n \ln{n})$，那么$\| \hat{N} - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$作为$n \to \infty$ ;如果$T_n = ω(n)$，一个稀疏的政权，那么$\| \hat{N}_Γ- \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$。在这里，$\hat{N}_Γ$是一个正则化，它将对应于跳转到最常访问的状态和从最常访问的状态跳转到最常访问的状态的条目清零。总之，这确定了BMC的顺序为$Θ_{\mathbb{P}}(\sqrt{T_n/n})$。

英文摘要：

This paper quantifies the asymptotic order of the largest singular value of a centered random matrix built from the path of a Block Markov Chain (BMC). In a BMC there are $n$ labeled states, each state is associated to one of $K$ clusters, and the probability of a jump depends only on the clusters of the origin and destination. Given a path $X_0, X_1, \ldots, X_{T_n}$ started from equilibrium, we construct a random matrix $\hat{N}$ that records the number of transitions between each pair of states. We prove that if $ω(n) = T_n = o(n^2)$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = Ω_{\mathbb{P}}(\sqrt{T_n/n})$. We also prove that if $T_n = Ω(n \ln{n})$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$ as $n \to \infty$; and if $T_n = ω(n)$, a sparser regime, then $\| \hat{N}_Γ- \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$. Here, $\hat{N}_Γ$ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is $Θ_{\mathbb{P}}(\sqrt{T_n/n})$ for BMCs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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