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Near-Optimal Clustering in Mixture of Markov Chains

Junghyun Lee, Yassir Jedra, Alexandre Proutière, Se-Young Yun

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 $S$. 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.

Near-Optimal Clustering in Mixture of Markov Chains

Abstract

We study the problem of clustering trajectories of length , each generated by one of K unknown ergodic Markov chains over a finite state space of size . 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 and . Preliminary experiments support our approach, and we conclude with discussions of its limitations and extensions.

Paper Structure

This paper contains 74 sections, 22 theorems, 127 equations, 6 figures.

Table of Contents

  1. INTRODUCTION
  2. Contributions.
  3. Notation.
  4. PROBLEM SETTING
  5. Mixture of Markov chains (MMCs).
  6. Learner's Objective.
  7. FUNDAMENTAL LOWER BOUND
  8. Asymptotically Exact Recovery.
  9. Stationary Divergence.
  10. TWO-STAGE ALGORITHM
  11. Stage I. Initial Spectral Clustering
  12. $L$-Embedding.
  13. Algorithm Design.
  14. Theoretical Analysis.
  15. Towards a Parameter-Free Algorithm.
  16. ...and 59 more sections

Key Result

theorem 1

Error Rate Lower Boundlower-bound Let $(\varepsilon, \delta) \in [0, 1] \times (0, 1/2]$. Then, a necessary condition for the existence of a $(\varepsilon, \beta, \delta)$-locally stable algorithm at $\Phi_T := (({\mathcal{M}}^{(k)})_{k \in [K]}, f, T)$ with $\beta \geq 2\sqrt{2} \varepsilon$ is as For $(\varepsilon, \beta)$-asymptotically locally stable algorithm with the same $\beta$ as above a

Figures (6)

  • Figure 1: Clustering error rates of our and kausik2023mixture's algorithms on Cyclic-Bump MMC.
  • Figure 2: (Ablation #1) Clustering error on the synthetic MMC instance across EM iterations (up to 10). Panels vary across $T \in \{100, 200, \cdots, 600\}$; the x-axis varies across $H \in \{10, 20, \cdots, 50\}$. Darker red curves indicate more EM iterations; "Oracle" and "Stage I" are shown for reference.
  • Figure 3: (Ablation #2) Impact of $S$ and $T$ on the resulting clustering error and runtime.
  • Figure 4: (Ablation #3) Clustering error of the unknown-$K$ variant on the synthetic MMC instance across different $(c_1, c_2)$ settings. Panels vary across $c_1 \in \{10^{-3}, 10^{-2}, 10^{-1}\}$ (columns) and trajectory length $H$ (rows: top for $H \leq 1000$, bottom for $H \geq 1000$). Curves indicate different $c_2$ values; Stage I and Stage I+II are shown separately.
  • Figure 5: (Ablation #4) Clustering error on another synthetic MMC instances with varying $\gamma_\mathrm{ps}$'s, over $H \in \{100, 200, \cdots, 2000\}$.
  • ...and 1 more figures

Theorems & Definitions (43)

  • definition 1
  • theorem 1
  • proof : Proof sketch
  • Remark 1: Chernoff Information
  • Remark 2: Comparison to wang2023divergence
  • definition 2
  • Remark 3
  • Proposition 4.1
  • proof
  • theorem 2
  • ...and 33 more