TY - GEN
T1 - Singular value decomposition of operators on reproducing kernel Hilbert spaces
AU - Mollenhauer, Mattes
AU - Schuster, Ingmar
AU - Klus, Stefan
AU - Schütte, Christof
N1 - Funding Information:
Acknowledgements. M. M., S. K., and C. S were funded by Deutsche Forschungs-gemeinschaft (DFG) through grant CRC 1114 (Scaling Cascades in Complex Systems, project ID: 235221301) and through Germany’s Excellence Strategy (MATH+: The Berlin Mathematics Research Center, EXC-2046/1, project ID: 390685689). We would like to thank Ilja Klebanov for proofreading the manuscript and valuable suggestions for improvements.
Publisher Copyright:
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020.
PY - 2020
Y1 - 2020
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85090253268&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-51264-4_5
DO - 10.1007/978-3-030-51264-4_5
M3 - Conference contribution
AN - SCOPUS:85090253268
SN - 9783030512637
T3 - Studies in Systems, Decision and Control
SP - 109
EP - 131
BT - Advances in Dynamics, Optimization and Computation. SON 2020
PB - Springer
ER -