Title: Spectral State Compression of Markov Processes,
Abstract: Model reduction of the Markov process is a basic problem in modeling
state-transition systems. Motivated by the state aggregation approach rooted in
control theory, we study the statistical state compression of a finite-state
Markov chain from empirical trajectories. Through the lens of spectral
decomposition, we study the rank and features of Markov processes, as well as
properties like representability, aggregatability, and lumpability. We develop
a class of spectral state compression methods for three tasks: (1) estimate the
transition matrix of a low-rank Markov model, (2) estimate the leading subspace
spanned by Markov features, and (3) recover latent structures of the state
space like state aggregation and lumpable partition. The proposed methods
provide an unsupervised learning framework for identifying Markov features and
clustering states. We provide upper bounds for the estimation errors and nearly
matching minimax lower bounds. Numerical studies are performed on synthetic
data and a dataset of New York City taxi trips.