Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups
This work provides a more efficient and robust method for modeling real-world random processes, though it appears incremental as it builds on existing spectral decomposition techniques.
The paper tackles the problem of learning continuous Markov semigroups by addressing the computational expense and limited scope of existing methods, achieving a reduction in computational complexity from quadratic to linear in the state dimension while ensuring accurate eigenvalue learning across time-lag variations.
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.