$L_2$-norm sampling discretization and recovery of functions from RKHS with finite trace

TL;DR

This paper investigates $L_2$-norm sampling discretization and recovery of functions in RKHS with finite trace, providing error estimates and convergence rates under minimal assumptions, including non-Mercer kernels.

Contribution

It offers new error bounds and convergence rates for sampling recovery in RKHS with finite trace, extending to non-separable and non-Mercer kernels without additional assumptions.

Findings

Provides concrete worst-case error estimates with explicit constants.

Achieves polynomial decay of failure probability with sample size.

Shows improved convergence rates under separability assumption.

Abstract

In this paper we study $L_2$-norm sampling discretization and sampling recovery of complex-valued functions in RKHS on $D \subset \R^d$ based on random function samples. We only assume the finite trace of the kernel (Hilbert-Schmidt embedding into $L_2$) and provide several concrete estimates with precise constants for the corresponding worst-case errors. In general, our analysis does not need any additional assumptions and also includes the case of non-Mercer kernels and also non-separable RKHS. The fail probability is controlled and decays polynomially in $n$, the number of samples. Under the mild additional assumption of separability we observe improved rates of convergence related to the decay of the singular values. Our main tool is a spectral norm concentration inequality for infinite complex random matrices with independent rows complementing earlier results by Rudelson, Mendelson, Pajor, Oliveira and Rauhut.