A sharp upper bound for sampling numbers in L2

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 t...

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Vydáno v:Applied and computational harmonic analysis Ročník 63; s. 113 - 134
Hlavní autoři: Dolbeault, Matthieu, Krieg, David, Ullrich, Mario
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.03.2023
Témata:
[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
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Shrnutí: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.
ISSN:1063-5203
1096-603X
DOI:10.1016/j.acha.2022.12.001