Learning Markov Models Via Low-Rank Optimization
Taming high-dimensional Markov models In “Learning Markov models via low-rank optimization”, Z. Zhu, X. Li, M. Wang, and A. Zhang focus on learning a high-dimensional Markov model with low-dimensional latent structure from a single trajectory of states. To overcome the curse of high dimensions, the authors propose to equip the standard MLE (maximum-likelihood estimation) with either nuclear norm regularization or rank constraint. They show that both approaches can estimate the full transition matrix accurately using a trajectory of length that is merely proportional to the number of states. To solve the rank-constrained MLE, which is a nonconvex problem, the authors develop a new DC (difference) programming algorithm. Finally, they apply the proposed methods to analyze taxi trips on the Manhattan island and partition the island based on the destination preference of customers; this partition can help balance supply and demand of taxi service and optimize the allocation of traffic resources.