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.

翻译：非线性动力学系统可通过对应的Koopman算子得到简洁描述，该算子的作用使系统的每个可观测量随时间向前演化。现有多种算法可从数据中学习Koopman算子及其谱分解。本文首次给出Koopman特征值与特征函数的非渐近学习界。我们专注于时间反演不变随机动力学系统，包括重要的Langevin动力学实例。分析了两种主流估计方法：扩展动态模态分解（EDMD）与降秩回归（RRR）。我们的关键结论依赖于算子范数误差的新型极小极大估计界——该结果本身可能具有独立研究价值。谱学习界受算子范数误差与估计特征函数的新型度量畸变泛函的联合控制。理论界表明：EDMD与RRR具有相近的方差，但EDMD受更大的偏差影响，这可能削弱其学习速率。我们的结果为虚假特征值的涌现提供了新解释——该问题在实证中广为人知。数值实验阐明了这些理论界在实践中的含义。