Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
This work provides a novel method for global analysis of dynamical systems, with potential applications in molecular dynamics, video, and text data, but it is incremental as it builds on existing kernel and operator theories.
The paper tackles the problem of analyzing complex dynamical systems by extending transfer operator theory to reproducing kernel Hilbert spaces, showing that these operators relate to conditional mean embeddings and enabling application to any domain with a kernel similarity measure.
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 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 in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to 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 these methods can be applied to any domain where a similarity measure given by a kernel is available. 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.