Estimating Koopman operators with sketching to provably learn large scale dynamical systems
This work addresses computational efficiency for researchers and practitioners analyzing complex dynamical systems, representing an incremental improvement over existing kernel-based estimators.
The paper tackles the challenge of scaling kernel-based Koopman operator estimators for large-scale dynamical systems by introducing random projections (sketching), resulting in estimators that retain the accuracy of existing methods like PCR or RRR while being much faster, as shown in experiments on synthetic and molecular dynamics datasets.
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.