Irregular Sampling of High-Dimensional Functions in Reproducing Kernel Hilbert Spaces
Armin Iske, Lennart Ohlsen
TL;DR
This work addresses the challenge of reconstructing high-dimensional functions in reproducing kernel Hilbert spaces (RKHS) from irregular samples taken at determining sequences. It develops sampling formulas of the form $f=\sum_{k\in\mathbb{N}} f(x_k) L_k$, with $L_k$ forming a dual Riesz basis to the kernel functions $K_{x_k}$, and uses the infinite Gram matrix $A_{K,X}$ to establish stable representations. The authors extend the framework to tensor-product RKHS, showing that tensor-product kernels yield composing determining samples and dual bases as tensor products, thereby reducing computational complexity for high-dimensional sampling. These results provide a rigorous, scalable method for high-dimensional kernel-based approximation from irregular samples, with potential impact on kernel regression, multivariate approximation, and related numerical analysis tasks. The approach hinges on PD kernels, Riesz theory, and the tensor-product structure to achieve efficient representations in high dimensions.
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.