Estimation of Markov Chain via Rank-Constrained Likelihood

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Xudong Li, Mengdi Wang, Anru Zhang ;
Proceedings of the 35th International Conference on Machine Learning, PMLR 80:3033-3042, 2018.

Abstract

This paper studies the estimation of low-rank Markov chains from empirical trajectories. We propose a non-convex estimator based on rank-constrained likelihood maximization. Statistical upper bounds are provided for the Kullback-Leiber divergence and the $\ell_2$ risk between the estimator and the true transition matrix. The estimator reveals a compressed state space of the Markov chain. We also develop a novel DC (difference of convex function) programming algorithm to tackle the rank-constrained non-smooth optimization problem. Convergence results are established. Experiments show that the proposed estimator achieves better empirical performance than other popular approaches.

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