Abstract
We provide a new upper bound for sampling numbers (g_n)_{nin mathbb {N}} associated with the compact embedding of a separable reproducing kernel Hilbert space into the space of square integrable functions. There are universal constants C,c>0 (which are specified in the paper) such that gn2≤Clog(n)n∑k≥⌊cn⌋σk2,n≥2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$\\end{document}where (sigma _k)_{kin mathbb {N}} is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding mathrm {Id}:H(K) rightarrow L_2(D,varrho _D). The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver’s conjecture, which was shown to be equivalent to the Kadison–Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of H^s_{text {mix}}(mathbb {T}^d) in L_2(mathbb {T}^d) with s>1/2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1/2}, \\end{aligned}$$\\end{document}which improves on very recent results by shortening the gap between upper and lower bound to sqrt{log (n)}. The result implies that for dimensions d>2 any sparse grid sampling recovery method does not perform asymptotically optimal.
Highlights
We study a well-known problem on the optimal recovery of multivariate functions from n function samples
The functions are modeled as elements from a separable reproducing kernel Hilbert space H (K ) of functions on a set D ⊂ Rd with finite trace kernel K (·, ·), i.e., tr(K ):= K (x, x)d D(x) < ∞
We start with a concentration inequality for the spectral norm of a matrix of type (12)
Summary
We study a well-known problem on the optimal recovery of multivariate functions from n function samples. We emphasize that in general, the square-summability of the singular numbers (σk )k∈N is not implied by the compactness of the embedding IdK , D. This is one reason why we need the additional assumption of a finite trace kernel (2) (or a Hilbert–Schmidt embedding). Given a measurable subset D ⊂ Rd and a measure , we denote with L2(D, ) the space of all square integrable complex-valued functions (equivalence classes) on D with D | f (x)|2 d (x) < ∞.
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