  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.
