# Estimation of Markov Chain via Rank-Constrained Likelihood

arXiv: https://arxiv.org/abs/1804.00795 · HTML: https://mlsanity.com/paper/1804.00795/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Apr 2018.

## Built on by (most cited first, 3 of 3)

- [From Self-Attention to Markov Models: Unveiling the Dynamics of Generative Transformers](https://mlsanity.com/paper/2402.13512/) (2024)
- [Spectral Entry-wise Matrix Estimation for Low-Rank Reinforcement Learning](https://mlsanity.com/paper/2310.06793/) (2023)
- [Low-Rank Tensors for Multi-Dimensional Markov Models](https://mlsanity.com/paper/2411.02098/) (2024)

## Abstract (arXiv)

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 ℓ2 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.