Abstract
In this paper we study L_2-norm sampling discretization and sampling recovery of complex-valued functions in RKHS on D subset mathbb {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.
Highlights
This paper can be seen as a continuation of [11,13]
We are interested in the sampling discretization of the squared L2-norm of such functions using n random nodes
The functions are modeled as elements from some reproducing kernel Hilbert space H (K ), which is supposed to be compactly embedded into L2(D, D)
Summary
This paper can be seen as a continuation of [11,13]. We study the reconstruction of complex-valued multivariate functions on a domain D ⊂ Rd from values at the (randomly sampled) nodes X := Xn) ∈ Dn via weighted least squares algorithms. We are interested in the sampling discretization of the squared L2-norm of such functions using n random nodes. Both problems recently gained substantial interest, see [11,13,14,25,26,27,29], and are strongly related as we know from Wasilkowski
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