L2-norm sampling discretization and recovery of functions from RKHS with finite trace

• Published in: Sampling theory, signal processing, and data analysis Vol. 19; no. 2
• Main Authors: Moeller, Moritz, Ullrich, Tino
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2021
• Subjects: Abstract Harmonic Analysis; Machine Learning; Mathematics; Mathematics and Statistics; Original Article; Signal,Image and Speech Processing; 68Q25; 41A25; Random sampling; Discretization; 41A63; Marcinkiewicz–Zygmund inequalities; 41A60; Least squares approximation; Spectral norm concentration; 94A20
• ISSN: 2730-5716; 2730-5724
• DOI: 10.1007/s43670-021-00013-3

In this paper we study L 2 -norm sampling discretization and sampling recovery of complex-valued functions in RKHS on D ⊂ 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.