Non-linear 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 from data is enabled by a number of algorithms. In this work we present nonasymptotic learning bounds for the Koopman eigenvalues and eigenfunctions estimated by two popular algorithms: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank Regression (RRR). We focus on time-reversal-invariant Markov chains, implying that the Koopman operator is self-adjoint. This includes important examples of stochastic dynamical systems, notably Langevin dynamics. Our spectral learning bounds are driven by the simultaneous control of the operator norm risk of the estimators and a metric distortion associated to the corresponding eigenfunctions. Our analysis indicates that both algorithms have similar variance, but EDMD suffers from a larger bias which might be detrimental to its learning rate. We further argue that a large metric distortion may lead to spurious eigenvalues, a phenomenon which has been empirically observed, and note that metric distortion can be estimated from data. Numerical experiments complement the theoretical findings.

翻译：相关 Koopman 操作员可以方便地描述非线性动态系统, 其动作会及时使该系统的每个可观测值发生演变。 从数据中学习 Koopman 操作员学习 库普曼 操作员是由若干算法促成的。 在这项工作中, 我们展示了由两种流行算法( 扩展动态模式分解( EDMD) 和 降级递减( RRRR) 估计的非线性学习边框 。 我们的分析表明, 这两种算法都存在类似的差异, 但是 EDMD 有更大的偏差, 这可能会损害到它的学习速度。 我们进一步论证, 大型的计量扭曲可能会导致刺激性神经性动态系统, 特别是兰格文动态。 我们的光谱学习边框是同时控制 Koopman 的操作员规范风险 和 与对应的脑功能相关的度扭曲 。 我们的分析表明, 这两种算法都有类似的差异, 但 EDMD 都有更大的偏差, 这可能会损害它的学习速度。 我们进一步论证说, 大型的参数变形变形可能会导致刺激性模拟价值, 。 模型的模型可以观察 。