Spectral norm bounds for block Markov chain random matrices

TL;DR

This paper establishes the asymptotic order of the largest singular value of a centered random matrix derived from a Block Markov Chain, revealing its dependence on the number of steps and states.

Contribution

It provides tight bounds on the spectral norm of the matrix constructed from BMC paths, including regularization techniques for sparse regimes, advancing understanding of spectral properties in Markov models.

Findings

Spectral norm scales as _{\u2014}(\u221a{T_n/n}) under certain conditions.

Regularization effectively controls the spectral norm in sparse regimes.

Results unify bounds for different regimes of the number of steps T_n.

Abstract

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 $\omega(n) = T_n = o(n^2)$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = \Omega_{\mathbb{P}}(\sqrt{T_n/n})$. We also prove that if $T_n = \Omega(n \ln{n})$, then $\| \hat{N} - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$ as $n \to \infty$; and if $T_n = \omega(n)$, a sparser regime, then $\| \hat{N}_\Gamma - \mathbb{E}[\hat{N}] \| = O_{\mathbb{P}}(\sqrt{T_n/n})$. Here, $\hat{N}_{\Gamma}$ is a regularization that zeroes out entries corresponding to jumps to and from most-often visited states. Together this establishes that the order is $\Theta_{\mathbb{P}}(\sqrt{T_n/n})$ for BMCs.