2,637 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
- …