# Estimating Koopman operators with sketching to provably learn large scale dynamical systems arXiv: https://arxiv.org/abs/2306.04520 ยท HTML: https://mlsanity.com/paper/2306.04520/ ## Key facts - Cited by 15 papers in the map (data through 14 Oct 2026). - Builds on 24 of its 70 references that are in the map. - First posted on arXiv 7 Jun 2023. ## Builds on (most cited first, 24 of 24) - [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017) - [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016) - [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021) - [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017) - [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017) - [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017) - [Less is More: Nystr"om Computational Regularization](https://mlsanity.com/paper/1507.04717/) (2015) - [Kernel methods through the roof: handling billions of points efficiently](https://mlsanity.com/paper/2006.10350/) (2020) - [FALKON: An Optimal Large Scale Kernel Method](https://mlsanity.com/paper/1705.10958/) (2017) - [Revisiting the Nystrom Method for Improved Large-Scale Machine Learning](https://mlsanity.com/paper/1303.1849/) (2013) - [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022) - [Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics](https://mlsanity.com/paper/1710.11239/) (2017) - [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022) - [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017) - [On Fast Leverage Score Sampling and Optimal Learning](https://mlsanity.com/paper/1810.13258/) (2018) - [A General Framework for Consistent Structured Prediction with Implicit Loss Embeddings](https://mlsanity.com/paper/2002.05424/) (2020) - [A Consistent Regularization Approach for Structured Prediction](https://mlsanity.com/paper/1605.07588/) (2016) - [Optimal Convergence for Distributed Learning with Stochastic Gradient Methods and Spectral Algorithms](https://mlsanity.com/paper/1801.07226/) (2018) - [Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)](https://mlsanity.com/paper/2106.01510/) (2021) - [Nystr"om Kernel Mean Embeddings](https://mlsanity.com/paper/2201.13055/) (2022) - [Learning to Forecast Dynamical Systems from Streaming Data](https://mlsanity.com/paper/2109.09703/) (2021) - [Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1807.09331/) (2018) - [Toward Large Kernel Models](https://mlsanity.com/paper/2302.02605/) (2023) - [Sketch In, Sketch Out: Accelerating both Learning and Inference for Structured Prediction with Kernels](https://mlsanity.com/paper/2302.10128/) (2023) ## Built on by (most cited first, 15 of 15) - [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023) - [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023) - [Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces](https://mlsanity.com/paper/2403.02524/) (2024) - [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024) - [Operator World Models for Reinforcement Learning](https://mlsanity.com/paper/2406.19861/) (2024) - [Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2505.18671/) (2025) - [Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om method](https://mlsanity.com/paper/2403.02811/) (2024) - [Deep Sketched Output Kernel Regression for Structured Prediction](https://mlsanity.com/paper/2406.09253/) (2024) - [Data-Driven Stochastic Optimal Control in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2407.16407/) (2024) - [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025) - [End-to-end guarantees for indirect data-driven control of bilinear systems with finite stochastic data](https://mlsanity.com/paper/2409.18010/) (2024) - [kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning](https://mlsanity.com/paper/2512.21409/) (2025) - [Koopman Subspace Pruning in Reproducing Kernel Hilbert Spaces via Principal Vectors](https://mlsanity.com/paper/2604.01459/) (2026) - [AdaKoop: Efficient Modeling of Nonlinear Dynamics from Nonstationary Data Streams with Koopman Operator Regression](https://mlsanity.com/paper/2606.04930/) (2026) - [Learning Metastable Dynamics](https://mlsanity.com/paper/2609.14712/) (2026) ## Abstract (arXiv) The theory of Koopman operators allows to deploy non-parametric machine learning algorithms to predict and analyze complex dynamical systems. Estimators such as principal component regression (PCR) or reduced rank regression (RRR) in kernel spaces can be shown to provably learn Koopman operators from finite empirical observations of the system's time evolution. Scaling these approaches to very long trajectories is a challenge and requires introducing suitable approximations to make computations feasible. In this paper, we boost the efficiency of different kernel-based Koopman operator estimators using random projections (sketching). We derive, implement and test the new "sketched" estimators with extensive experiments on synthetic and large-scale molecular dynamics datasets. Further, we establish non asymptotic error bounds giving a sharp characterization of the trade-offs between statistical learning rates and computational efficiency. Our empirical and theoretical analysis shows that the proposed estimators provide a sound and efficient way to learn large scale dynamical systems. In particular our experiments indicate that the proposed estimators retain the same accuracy of PCR or RRR, while being much faster.