A sharp upper bound for sampling numbers in $L_{2}$

TL;DR

This paper establishes a universal upper bound relating sampling numbers and Kolmogorov widths for function classes in $L_2$, using advanced techniques including the solution to the Kadison-Singer problem.

Contribution

It provides a sharp, universal upper bound for sampling numbers in $L_2$ for classes like RKHS unit balls, extending to general compact classes and demonstrating the bounds' sharpness.

Findings

Established a universal constant for the upper bound.

Derived bounds for classes of continuous functions.

Proved the bounds are sharp with matching examples.

Abstract

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 \[ g_{cn}(F)^2 \,\le\, \frac{1}{n}\sum_{k\geq n} d_k(F)^2, \] 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.