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.

本文提出一种基于随机投影的非参数机器学习算法，可更高效的学习 Koopman 算子，以分析和预测复杂的动态系统，实验证明提出的估计器在保持相同精确度的情况下比 PCR 或 RRR 更快。