Clustering in Mixtures of Ergodic Markov Chains: Fundamental Limit and Two-Stage Algorithm

We study the problem of clustering $T$ trajectories of length $H$, each generated by one of $K$ unknown ergodic Markov chains over a finite state space of size $S$. We derive an instance-dependent, high-probability lower bound on the clustering error rate, governed by the stationary-weighted KL divergence between transition kernels. We then propose a two-stage algorithm: Stage I applies spectral clustering via a new injective Euclidean embedding for ergodic Markov chains, a contribution of independent interest enabling sharp concentration results; Stage II refines clusters with a single likelihood-based reassignment step. We prove that our algorithm achieves near-optimal clustering error with high probability under reasonable requirements on $T$ and $H$. Preliminary experiments support our approach, and we conclude with discussions of its limitations and extensions.