# Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

arXiv: https://arxiv.org/abs/1807.09331 · HTML: https://mlsanity.com/paper/1807.09331/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2018.

## Built on by (most cited first, 14 of 14)

- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023)
- [Dynamical systems and complex networks: A Koopman operator perspective](https://mlsanity.com/paper/2405.08940/) (2024)
- [Spatio-temporal point processes with deep non-stationary kernels](https://mlsanity.com/paper/2211.11179/) (2022)
- [Variational Search Distributions](https://mlsanity.com/paper/2409.06142/) (2024)
- [Symmetric and antisymmetric kernels for machine learning problems in quantum physics and chemistry](https://mlsanity.com/paper/2103.17233/) (2021)
- [A randomized algorithm to solve reduced rank operator regression](https://mlsanity.com/paper/2312.17348/) (2023)
- [Spectral Representation-based Reinforcement Learning](https://mlsanity.com/paper/2512.15036/) (2025)
- [Learning Transfer Operators by Kernel Density Estimation](https://mlsanity.com/paper/2210.03124/) (2022)
- [Learning Stochastic Nonlinear Dynamics with Embedded Latent Transfer Operators](https://mlsanity.com/paper/2501.02721/) (2025)
- [Learning graphons from data: Random walks, transfer operators, and spectral clustering](https://mlsanity.com/paper/2507.18147/) (2025)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)
- [Deep spatio-temporal point processes: Advances and new directions](https://mlsanity.com/paper/2504.06364/) (2025)
- [Functional dynamic mode decomposition: Learning infinite-dimensional systems from data](https://mlsanity.com/paper/2609.29159/) (2026)

## Abstract (arXiv)

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.