[Paper Review] Spectral State Compression of Markov Processes

Anru R. Zhang, Mengdi Wang|arXiv (Cornell University)|Feb 8, 2018

Fault Detection and Control Systems63 references3 citations

TL;DR

This paper proposes a spectral method for state compression of finite-state Markov chains using low-rank approximation of the transition matrix, leveraging spectral decomposition to estimate leading feature subspaces and recover latent structures like state aggregation or lumpable partitions. It establishes minimax-optimal statistical guarantees for transition matrix estimation and subspace recovery under finite-sample conditions.

ABSTRACT

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.

Motivation & Objective

• To develop an unsupervised, data-driven method for compressing large discrete-state Markov chains using spectral decomposition.
• To estimate the low-rank structure of the transition kernel from empirical trajectories without prior knowledge of meta-states or partitions.
• To provide sharp statistical guarantees—upper and lower bounds—for transition matrix estimation and subspace recovery under finite-sample settings.
• To recover latent structures such as state aggregation and lumpable partitions with statistical confidence.
• To unify the concepts of aggregability, lumpability, and low-rank decomposition through spectral analysis of Markov processes.

Proposed method

• The method uses spectral decomposition of the transition matrix to represent it as a sum of rank-one components: $\mathbb{P}(X_{t+1}|X_t) \approx \sum_{k=1}^r f_k(X_t)g_k(X_{t+1})$, where $f_k$ and $g_k$ are feature functions.
• It estimates the leading $r$-dimensional subspace spanned by the dominant left and right singular vectors of the transition matrix via empirical covariance estimation from trajectories.
• A spectral truncation procedure is applied to the empirical transition matrix to obtain a low-rank approximation, minimizing total variation error.
• The method incorporates mixing time and stationary distribution assumptions to control estimation bias and variance.
• Freedman’s inequality is used to derive high-probability concentration bounds on the estimation error of the empirical feature matrix.
• Theoretical analysis connects spectral properties to structural concepts like aggregability and lumpability, enabling recovery of latent partitions.

Experimental results

Research questions

• RQ1 Can we estimate the low-rank structure of a Markov transition matrix from finite-length trajectories with sharp statistical error bounds?
• RQ2 How well can spectral methods recover the leading feature subspace of a Markov process when the true rank is unknown?
• RQ3 What is the minimax optimal rate for estimating a low-rank Markov model from empirical data?
• RQ4 Under what conditions can spectral methods recover state aggregation or lumpable partitions with statistical confidence?
• RQ5 How do spectral decomposition and singular vector estimation relate to the concepts of aggregability and lumpability in Markov chains?

Key findings

• The proposed spectral method achieves an upper bound on total variation error of $O\left(\sqrt{\frac{\pi_{\max} \delta^2 r \tau_* \log^2 n}{np}}\right)$ for transition matrix estimation, with high probability.
• A nearly-matching minimax lower bound is established, confirming the statistical optimality of the proposed method under the given model assumptions.
• For general Markov chains, the method recovers the leading Markov feature subspace with error bounded by $O\left(\sqrt{\frac{\pi_{\max} \delta^2 r \tau_* \log^2 n}{np}}\right)$ with high probability.
• In the case of aggregable or lumpable processes, the method can recover the underlying state partition with statistical consistency guarantees.
• Theoretical results are validated numerically on synthetic data and real-world taxi trip trajectories from New York City, demonstrating accurate low-rank approximation and structure recovery.
• The method achieves minimax-optimality in estimation error, confirming that no other method can achieve better finite-sample performance under the same assumptions.