3 research outputs found
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many
statistics and machine learning applications ranging from support vector
machines to Gaussian processes and kernel embeddings of distributions.
Operators acting on such spaces are, for instance, required to embed
conditional probability distributions in order to implement the kernel Bayes
rule and build sequential data models. It was recently shown that transfer
operators such as the Perron-Frobenius or Koopman operator can also be
approximated in a similar fashion using covariance and cross-covariance
operators and that eigenfunctions of these operators can be obtained by solving
associated matrix eigenvalue problems. The goal of this paper is to provide a
solid functional analytic foundation for the eigenvalue decomposition of RKHS
operators and to extend the approach to the singular value decomposition. The
results are illustrated with simple guiding examples
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
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