Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups

TL;DR

This paper introduces a low-complexity, Laplace transform-based method for learning continuous Markov semigroups that is robust to small time-lags and applicable to a broad class of processes.

Contribution

It proposes a novel approach using the resolvent of the infinitesimal generator, reducing computational complexity and providing guarantees for transfer operator recovery.

Findings

Accurately learns eigenvalues for small time-lags

Reduces computational complexity from quadratic to linear

Applicable to a broader class of Markov processes

Abstract

Markov processes serve as a universal model for many real-world random processes. This paper presents a data-driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector-valued regression and Hilbert-Schmidt operator analysis. Existing techniques, including physics-informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time-lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time-lag variations, ensuring accurate eigenvalue learning even for small time-lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments.