A Note on Projection-Based Recovery of Clusters in Markov Chains

TL;DR

This paper presents a simple spectral projection algorithm for recovering clusters in perturbed Markov chains, providing bounds on the perturbation size for exact and approximate cluster recovery.

Contribution

It introduces a projection-based method for cluster recovery in Markov chains under perturbations, with explicit bounds on the perturbation magnitude for success.

Findings

Exact cluster recovery when perturbation is small enough, proportional to the spectral gap and inverse square root of largest cluster size.

Approximate recovery of a single cluster under milder perturbation conditions.

The method relies on singular value decomposition of the matrix I - T(x).

Abstract

Let $T_0$ be the transition matrix of a purely clustered Markov chain, i.e. a direct sum of $k \geq 2$ irreducible stochastic matrices. Given a perturbation $T(x) = T_0 + xE$ of $T_0$ such that $T(x)$ is also stochastic, how small must $x$ be in order for us to recover the indices of the direct summands of $T_0$? We give a simple algorithm based on the orthogonal projection matrix onto the left or right singular subspace corresponding to the $k$ smallest singular values of $I - T(x)$ which allows for exact recovery all clusters when $x = O\left(\frac{\sigma_{n - k}}{||E||_2\sqrt{n_1}}\right)$ and approximate recovery of a single cluster when $x = O\left(\frac{\sigma_{n - k}}{||E||_2}\right)$, where $n_1$ is the size of the largest cluster and $\sigma_{n - k}$ the $(k + 1)$st smallest singular value of $T_0$.