mlsanityData to 7 Oct 2026

Sharp Spectral Rates for Koopman Operator Learning

arXiv 2302.02004 ↗First posted 3 Feb 2023cs.LG math.DScited by 35

Not on the map. Why?

Abstract (arXiv)

Nonlinear dynamical systems can be handily described by the associated Koopman operator, whose action evolves every observable of the system forward in time. Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms. In this work we present for the first time non-asymptotic learning bounds for the Koopman eigenvalues and eigenfunctions. We focus on time-reversal-invariant stochastic dynamical systems, including the important example of Langevin dynamics. We analyze two popular estimators: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank Regression (RRR). Our results critically hinge on novel {minimax} estimation bounds for the operator norm error, that may be of independent interest. Our spectral learning bounds are driven by the simultaneous control of the operator norm error and a novel metric distortion functional of the estimated eigenfunctions. The bounds indicates that both EDMD and RRR have similar variance, but EDMD suffers from a larger bias which might be detrimental to its learning rate. Our results shed new light on the emergence of spurious eigenvalues, an issue which is well known empirically. Numerical experiments illustrate the implications of the bounds in practice.

Key facts. Cited by 35 papers in the map (data through 7 Oct 2026). Builds on 11 of its 49 references that are in the map. First posted on arXiv 3 Feb 2023.

Builds on

11 of 49 references.

References · 11

  1. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  2. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  3. 2017Sobolev Norm Learning Rates for Regularized Least-Squares Algorithmcited by 113
  4. 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 66
  5. 2017Time-lagged autoencoders: Deep learning of slow collective variables for molecular kineticscited by 64
  6. 2017Variational approach for learning Markov processes from time series datacited by 51
  7. 2022Optimal Rates for Regularized Conditional Mean Embedding Learningcited by 50
  8. 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 44
  9. 2020A General Framework for Consistent Structured Prediction with Implicit Loss Embeddingscited by 35
  10. 2022Spectral Representation Learning for Conditional Moment Modelscited by 12
  11. 2016Low-Rank Dynamic Mode Decomposition: An Exact and Tractable Solutioncited by 2

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 11; 2025: 11; 2026: 10.

35 papers.

Sort:

By area:

  1. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  2. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
  3. 2023Consistent Long-Term Forecasting of Ergodic Dynamical Systemscited by 10
  4. 2024Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spacescited by 9
  5. 2024Learning the Infinitesimal Generator of Stochastic Diffusion Processescited by 9
  6. 2024Nonparametric Control Koopman Operatorscited by 7
  7. 2024Operator World Models for Reinforcement Learningcited by 7
  8. 2024Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroupscited by 7
  9. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 7
  10. 2025Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systemscited by 7
  11. 2024From Biased to Unbiased Dynamics: An Infinitesimal Generator Approachcited by 6
  12. 2025Koopman-Equivariant Gaussian Processescited by 6
23 more
  1. 2025Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systemscited by 5
  2. 2025Demystifying Spectral Feature Learning for Instrumental Variable Regressioncited by 4
  3. 2025Information Shapes Koopman Representationcited by 3
  4. 2024Data-Driven Stochastic Optimal Control in Reproducing Kernel Hilbert Spacescited by 2
  5. 2025Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operatorscited by 2
  6. 2025A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systemscited by 2
  7. 2024Nonparametric Sparse Online Learning of the Koopman Operatorcited by 1
  8. 2024Kernel-Based Optimal Control: An Infinitesimal Generator Approachcited by 1
  9. 2025kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learningcited by 1
  10. 2026Toeplitz Based Spectral Methods for Data-driven Dynamical Systemscited by 1
  11. 2026Koopman operator theory: fundamentals, control, and applicationscited by 1
  12. 2023Koopman Learning with Episodic Memorycited by 0
  13. 2025Nonparametric Sparse Online Learning of the Koopman Operatorcited by 0
  14. 2025Predicting the Energy Landscape of Stochastic Dynamical System via Physics-informed Self-supervised Learningcited by 0
  15. 2025Conformal Online Learning of Deep Koopman Linear Embeddingscited by 0
  16. 2026Geometric Dictionary Learning of Dynamical Systems with Optimal Transportcited by 0
  17. 2026Finding Koopman Invariant Subspaces via Personalized PageRankcited by 0
  18. 2026AdaKoop: Efficient Modeling of Nonlinear Dynamics from Nonstationary Data Streams with Koopman Operator Regressioncited by 0
  19. 2026Reward-free Pretraining for Reinforcement Learning via Occupancy Coverage Maximizationcited by 0
  20. 2026Learning to control switching nonlinear systems with Koopman operator regressioncited by 0
  21. 2026Demystifying Linear Operator Learning for Control Systemscited by 0
  22. 2026Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantificationcited by 0
  23. 2026Learning Conditional Expectation Operators via Functional Newton Updatescited by 0