Norm-optimal model order reduction in reproducing kernel Hilbert spaces
Abstract
We consider the problem of model order reduction (MOR) in reproducing kernel Hilbert spaces (RKHSs). We prove generalizations of well-known necessary interpolatory conditions for norm-optimal reduced-order models, originally stated for rational functions in H2 Hardy-Hilbert spaces. We construct an alternative to the iterative rational Krylov (IRKA) algorithm for RKHSs with differentiable kernels to obtain norm-optimal reduced-order models. Then, we introduce generalized Malmquist-Takenaka (GMT) functions that can be used to define orthonormal bases in an arbitrary RKHS. Using GMT functions, we propose unitary transformations between separable RKHSs. This allows us to extend our optimality conditions and subsequent algorithms to MOR problems posed in RKHSs with non-differentiable kernels. The potential and versatility of the proposed MOR framework is demonstrated in several numerical experiments. We perform norm-optimal MOR on various machine learning models such as support vector machines and deep ReLU networks. In addition, we show how the proposed MOR framework can be used to solve seemingly unrelated problems. In particular, we provide an example for fitting high degree polynomials to data and also obtain a FEM-like method for solving partial differential equations (PDEs) with an optimal kernel basis.
Related
- Multiplicative linear functionals on reproducing kernel Hilbert spaces
- A Framework for Supervised and Unsupervised Learning via Reproducing Kernel Hilbert Spaces
- Gaussian Processes with Sample Paths in Reproducing Kernel Banach Spaces
- The strong reproducing kernel thesis for Hankel operators and the Garsia norm
- A Multidimensional Hilbert-Type Integral Inequality with the General Nonhomogeneous Kernel and Multiple Lower Limit Function
BibTeX
@misc{dozsa2026normoptimal,
title = {{Norm-optimal model order reduction in reproducing kernel Hilbert spaces}},
author = {Tamás Dózsa and Andrea Angino and Rolf Krause and Matthias Voigt},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28224},
eprint = {2610.28224},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28224},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Norm-optimal model order reduction in reproducing kernel Hilbert spaces
AU - Tamás Dózsa
AU - Andrea Angino
AU - Rolf Krause
AU - Matthias Voigt
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28224
UR - https://arxiv.org/abs/2610.28224
AB - We consider the problem of model order reduction (MOR) in reproducing kernel Hilbert spaces (RKHSs). We prove generalizations of well-known necessary interpolatory conditions for norm-optimal reduced-order models, originally stated for rational functions in H_2 Hardy-Hilbert spaces. We construct an alternative to the iterative rational Krylov (IRKA) algorithm for RKHSs with differentiable kernels to obtain norm-optimal reduced-order models. Then, we introduce generalized Malmquist-Takenaka (GMT) functions that can be used to define orthonormal bases in an arbitrary RKHS. Using GMT functions, we propose unitary transformations between separable RKHSs. This allows us to extend our optimality conditions and subsequent algorithms to MOR problems posed in RKHSs with non-differentiable kernels. The potential and versatility of the proposed MOR framework is demonstrated in several numerical experiments. We perform norm-optimal MOR on various machine learning models such as support vector machines and deep ReLU networks. In addition, we show how the proposed MOR framework can be used to solve seemingly unrelated problems. In particular, we provide an example for fitting high degree polynomials to data and also obtain a FEM-like method for solving partial differential equations (PDEs) with an optimal kernel basis.
ER -