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Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups

ICML 2025 · Learning theory and algorithms › Kernels and regression theory · approach: Kernel regression under shift · 0 citations (OpenAlex)

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Proposes LaRRR, a Laplace-transform-based reduced-rank regression method that learns the resolvent (and hence spectral decomposition) of a Markov semigroup's infinitesimal generator directly from a single trajectory, with linear complexity in state dimension and guarantees valid at any time-lag.

Problem

Learning the spectral decomposition of the unbounded infinitesimal generator of a continuous Markov semigroup from trajectory data; transfer operator methods fail for small time-lags, and existing generator methods are limited in scope and scale quadratically or worse with state dimension.

Approach

Approximate the resolvent of the generator via the Laplace transform of transfer operators using trapezoid-rule quadrature over multiple time-lags, then apply vector-valued reduced rank regression in an RKHS; the resolvent is bounded so the estimation is stable as the time-lag shrinks, and the empirical aggregation reduces to a Toeplitz matrix product.

Key results

• On the 1D triple-well Langevin, LaRRR's error for lambda_2 stays stable as the time-lag decreases while the transfer operator RRR error explodes as Delta t -> 0 (Delta t = 0.005 shown).
• For the 2x2 non-normal OU process, primal LaRRR with 1000 random Fourier features achieves better eigenvalue estimates than TO RRR over ten trials at two time-discretizations.
• Dual LaRRR recovers two metastable states of alanine dipeptide (d=45 interatomic distances) consistent with molecular dynamics expert knowledge.

Measured effects reported in the paper

Surprising findings

• Tuning the shift parameter mu upward is prohibitive: large mu makes the resolvent's eigenvalues collapse to zero, so mu must be kept small for spectral estimation.

Limitations

• Theory assumes uniform sampling of the full state along a single trajectory; non-uniform sampling and partially observed systems are left open.
• Eigenvalue bounds for non-normal generators are multiplied by conditioning factors, and eigenvalue estimates can be highly variable under non-normality.
• Spectral learning rate is slower (n^{-1/3}) than the operator-norm rate, and the regularity parameters alpha, beta must be assumed.

Experimental setup

Datasets: simulated Ornstein-Uhlenbeck trajectories simulated 1D overdamped Langevin triple-well trajectories alanine dipeptide molecular dynamics trajectories

Benchmarks: Ornstein-Uhlenbeck 2x2 non-normal drift 1D triple-well Langevin alanine dipeptide metastable states

Metrics: relative eigenvalue error eigenfunction recovery quality

Baselines: transfer operator RRR physics-informed kernel regression (Dirichlet form) kernel generator learning with Dirichlet form

Builds on: transfer operator regression physics-informed kernel regression for generator learning vector-valued reduced rank regression Crouzeix's theorem

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