Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces

2025/04/18 by Iske, Armin, Ohlsen, Lennart
#FOS: Computer and information sciences #Information Theory (cs.IT) #Machine Learning (cs.LG)

paper · doi:10.48550/arxiv.2504.13543

Abstract

We develop sampling formulas for high-dimensional functions in reproducing kernel Hilbert spaces, where we rely on irregular samples that are taken at determining sequences of data points. We place particular emphasis on sampling formulas for tensor product kernels, where we show that determining irregular samples in lower dimensions can be composed to obtain a tensor of determining irregular samples in higher dimensions. This in turn reduces the computational complexity of sampling formulas for high-dimensional functions quite significantly.

Citations

• On the convergence of generalized kernel-based interpolation by greedy data selection algorithms
• On the convergence of generalized kernel-based interpolation by greedy data selection algorithms
• Product kernels are efficient and flexible tools for high-dimensional scattered data interpolation
• Dimensionality Reduction and Wasserstein Stability for Kernel Regression
• Understanding neural networks with reproducing kernel Banach spaces
• Kernel dimension reduction in regression

Related

• Nonuniform sampling, reproducing kernels, and the associated Hilbert spaces
• On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces
• Kernel interpolation in Sobolev spaces of hybrid regularity
• Sampling basis in reproducing kernel Banach spaces
• Breaking the Curse for Uniform Approximation in Hilbert Spaces via Monte Carlo Methods