A sharp upper bound for sampling numbers in $L_2$
Résumé
For a class $F$ of complex-valued functions on a set $D$, we denote by $g_n(F)$ its sampling numbers, i.e., the minimal worst-case error on $F$, measured in $L_2$, that can be achieved with a recovery algorithm based on $n$ function evaluations. We prove that there is a universal constant $c \in \mathbb N$ such that, if $F$ is the unit ball of a separable reproducing kernel Hilbert space, then $g_{cn}(F)^2 \leq \frac 1 n \sum_{k \geq n} d_k(F)^2$, where $d_k(F)$ are the Kolmogorov widths (or approximation numbers) of $F$ in $L_2$. We also obtain similar upper bounds for more general classes $F$, including all compact subsets of the space of continuous functions on a bounded domain $D \subset \mathbb R^d$, and show that these bounds are sharp by providing examples where the converse inequality holds up to a constant. The results rely on the solution to the Kadison-Singer problem, which we extend to the subsampling of a sum of infinite rank-one matrices.
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