Transfer operators such as the Perron–Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose kernel transfer operators, which extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.

中文翻译：

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再现核希尔伯特空间中转移算子的本征分解

Perron–Frobenius或Koopman运算符之类的传递运算符在复杂动力学系统的全局分析中起着重要作用。这些算子的本征函数可用于检测亚稳态集，将动力学投影到主要的慢速过程中或分离叠加的信号。我们建议内核转移运算符，将传递算子理论扩展到再现内核希尔伯特空间，并表明这些算子与条件分布的希尔伯特空间表示（称为条件均值嵌入）有关。所提出的用于计算这些内核转移算子经验估计的数值方法包含了现有的数据驱动方法，用于近似转移算子，例如扩展动态模式分解及其变体。所提出的基于内核的方法的一个主要优点是，它们可以应用于可以使用内核给出的相似性度量的任何领域。此外，我们提供了关于有限秩RKHS算子的本征分解的基本结果。