A New Upper Bound for Sampling Numbers

Abstract

We provide a new upper bound for sampling numbers (gn)n∈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,where (σk)k∈N is the sequence of singular numbers (approximation numbers) of the Hilbert–Schmidt embedding Id : H(K) → L2(D, ϱD). 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 Hmixs(Td) in L2(Td) with s> 1 / 2. We obtain the asymptotic bound gn≤Cs,dn-slog(n)(d-1)s+1/2,which improves on very recent results by shortening the gap between upper and lower bound to log(n). The result implies that for dimensions d> 2 any sparse grid sampling recovery method does not perform asymptotically optimal.