Estimation of Markov Chain via Rank-Constrained Likelihood
This work addresses the estimation of Markov chains for applications requiring compressed state spaces, but it appears incremental as it builds on existing rank-constrained methods with specific algorithmic improvements.
The paper tackles the problem of estimating low-rank Markov chains from empirical trajectories by proposing a non-convex estimator based on rank-constrained likelihood maximization, achieving better empirical performance than other popular approaches in experiments.
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.