A new upper bound for sampling numbers

We provide a new upper bound for sampling numbers $(gn){n\in \mathbb{N}}$ associated to 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 $$ g^2_n \leq \frac{C\log(n)}{n}\sum\limits{k\geq \lfloor cn \rfloor} \sigmak^2\quad,\quad n\geq 2\,, $$ where $(\sigmak){k\in \mathbb{N}}$ is the sequence of singular numbers (approximation numbers) of the Hilbert-Schmidt embedding $\text{Id}:H(K) \to L2(D,\varrhoD)$. 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 $L2(\mathbb{T}^d)$ with $s>1/2$. We obtain the asymptotic bound $$ gn \leq C_{s,d}n^{{-s}\log(n){(d-1)s+1/2}\,,} $$ which improves on very recent results by shortening the gap between upper and lower bound to $\sqrt{\log(n)}$.