Work

Noisy Nonlinear Information and Entropy Numbers

Year: 2026

Type: article

Abstract: It is impossible to recover a vector from $$\mathbb {R}^m$$ R m with less than m linear measurements, even if the measurements are chosen adaptively. Recently, it has been shown that one can recover vectors from $$\mathbb {R}^m$$ R m with arbitrary precision using only $$O(\log m)$$ O ( log m ) continuous (even Lipschitz) adaptive measurements, resulting in an exponential speed-up of continuous information compared to linear information for various approximation problems. In this note, we characterize the quality of optimal (dis-)continuous information that is disturbed by deterministic noise in terms of entropy numbers. This shows that in the presence of noise the potential gain of continuous over linear measurements is limited, but significant in some cases.

Source: Journal of Fourier Analysis and Applications

DOI: 10.1007/s00041-026-10249-z

Authors: David Krieg, Erich Novak, Leszek Plaskota, Mario Ullrich

Institutions: University of Passau, Friedrich Schiller University Jena, University of Warsaw, Johannes Kepler University of Linz

Language: English

Topic: Mathematical Analysis and Transform Methods

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,990

Funder: Friedrich-Schiller-Universität Jena