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Model reduction of Markov processes 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 discrete-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, aggregability, and lumpability. We develop spectral methods for estimating the transition matrix of a low-rank Markov model, estimating the leading subspace spanned by Markov features, and recovering latent structures like state aggregation and lumpable partition of the state space. We prove statistical 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.
We propose a deep generative Markov State Model (DeepGenMSM) learning framework for inference of metastable dynamical systems and prediction of trajectories. After unsupervised training on time series data, the model contains (i) a probabilistic enco
We formulate approximate Bayesian inference in non-conjugate temporal and spatio-temporal Gaussian process models as a simple parameter update rule applied during Kalman smoothing. This viewpoint encompasses most inference schemes, including expectat
Gaussian process regression (GPR) is a non-parametric Bayesian technique for interpolating or fitting data. The main barrier to further uptake of this powerful tool rests in the computational costs associated with the matrices which arise when dealin
New relations between ergodic rate, L_p convergence rates, and asymptotic behavior of tail probabilities for hitting times of a time homogeneous Markov process are established. For L_p convergence rates and related spectral and functional properties
As Gaussian processes are used to answer increasingly complex questions, analytic solutions become scarcer and scarcer. Monte Carlo methods act as a convenient bridge for connecting intractable mathematical expressions with actionable estimates via s