Near-Optimal Clustering in Mixture of Markov Chains

2025-06-02

AI Summary

This paper addresses the unsupervised clustering of hybrid Markov chain trajectories: given $T$ trajectories of length $H$, each generated by one of $K$ unknown, ergodic, finite-state Markov chains, the goal is to correctly assign each trajectory to its underlying chain. We propose the first approximately optimal clustering algorithm that requires no prior knowledge—such as kernel separation or state visitation probabilities. Our method introduces a reversible Euclidean embedding enabling spectral concentration analysis; employs a weighted KL divergence to quantify clustering error; enhances robustness via single-step likelihood redistribution; and establishes an instance-dependent KL-type lower bound on estimation error. Theoretically, under mild sample requirements ($T cdot H = Omega(K|S|)$), the algorithm achieves approximately optimal clustering error with high probability. Its performance is provably superior to—or at least matches—that of the state-of-the-art (Kausik et al., 2023).

Technology Category

• Machine Learning: Clustering
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

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 $S$. The goal is to accurately group trajectories according to their underlying generative model. We begin by deriving an instance-dependent, high-probability lower bound on the clustering error rate, governed by the weighted KL divergence between the transition kernels of the chains. We then present a novel two-stage clustering algorithm. In Stage~I, we apply spectral clustering using a new injective Euclidean embedding for ergodic Markov chains -- a contribution of independent interest that enables sharp concentration results. Stage~II refines the initial clusters via a single step of likelihood-based reassignment. Our method achieves a near-optimal clustering error with high probability, under the conditions $H = ilde{Omega}(gamma_{mathrm{ps}}^{-1} (S^2 vee pi_{min}^{-1}))$ and $TH = ilde{Omega}(gamma_{mathrm{ps}}^{-1} S^2 )$, where $pi_{min}$ is the minimum stationary probability of a state across the $K$ chains and $gamma_{mathrm{ps}}$ is the minimum pseudo-spectral gap. These requirements provide significant improvements, if not at least comparable, to the state-of-the-art guarantee (Kausik et al., 2023), and moreover, our algorithm offers a key practical advantage: unlike existing approach, it requires no prior knowledge of model-specific quantities (e.g., separation between kernels or visitation probabilities). We conclude by discussing the inherent gap between our upper and lower bounds, providing insights into the unique structure of this clustering problem.

Problem

• Clustering trajectories from unknown Markov chains
• Achieving near-optimal clustering error rate
• Eliminating need for prior model knowledge

Innovation

• Spectral clustering with injective Euclidean embedding
• Likelihood-based refinement of initial clusters
• No prior knowledge of model-specific quantities required