A sharp upper bound for sampling numbers in L2

• Published in: Applied and computational harmonic analysis Vol. 63; pp. 113 - 134
• Main Authors: Dolbeault, Matthieu, Krieg, David, Ullrich, Mario
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2023
• Subjects: [formula omitted]-approximation; Information-based complexity; Kadison-Singer; Least squares; Random matrices; Rate of convergence; Kadison-Singer; 41A45; 46B15; Random matrices; 41A25; Information-based complexity; Least squares; 60B20; L2-approximation; Rate of convergence; 46B09
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2022.12.001

For a class F of complex-valued functions on a set D, we denote by gn(F) its sampling numbers, i.e., the minimal worst-case error on F, measured in L2, that can be achieved with a recovery algorithm based on n function evaluations. We prove that there is a universal constant c∈N such that, if F is the unit ball of a separable reproducing kernel Hilbert space, thengcn(F)2≤1n∑k≥ndk(F)2, where dk(F) are the Kolmogorov widths (or approximation numbers) of F in L2. 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⊂Rd, 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.