Volume 56, Number 1, January-February 2022
|Page(s)||79 - 104|
|Published online||07 February 2022|
Convergence bounds for empirical nonlinear least-squares
Weierstrass Institute, Mohrenstrasse 39, D-10117 , Berlin, Germany
2 TU Berlin, Straße des 17 Juni 136, D-10623 , Berlin, Germany
* Corresponding author: firstname.lastname@example.org
Accepted: 20 October 2021
We consider best approximation problems in a nonlinear subset ℳ of a Banach space of functions (𝒱,∥•∥). The norm is assumed to be a generalization of the L 2-norm for which only a weighted Monte Carlo estimate ∥•∥n can be computed. The objective is to obtain an approximation v ∈ ℳ of an unknown function u ∈ 𝒱 by minimizing the empirical norm ∥u − v∥n. We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the specified nonlinear least squares setting. Several model classes are examined and the analytical statements about the RIP are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.
Mathematics Subject Classification: 62J02 / 41A25 / 41A65 / 41A30
Key words: Weighted nonlinear least squares / error analysis / convergence rates / weighted sparsity / tensor networks
© The authors. Published by EDP Sciences, SMAI 2022
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