Tractability of sampling recovery on unweighted function classes

TL;DR

The paper studies sampling recovery in $L_2$ for unweighted high-dimensional function classes and shows that nonlinearity enables tractability where linear methods fail. By embedding these classes in the Wiener algebra and employing a nonlinear reconstruction based on basis pursuit denoising, the authors derive a general complexity bound that depends on a bounded orthonormal system and projection error; this yields polynomial tractability under mild growth conditions. For concrete unweighted classes with absolutely convergent Fourier series, such as $F_d^{\log}$, Sobolev mixed spaces, and Hölder classes, the results obtain explicit $n(\varepsilon,\cdot, L_p)$ bounds demonstrating tractability in $L_p$ ($1\le p<\infty$) while linear algorithms remain intractable. This highlights the practical benefit of nonlinear sampling strategies in high-dimensional approximation and connects compressed sensing techniques to function approximation in unweighted settings.

Abstract

It is well-known that the problem of sampling recovery in the $L_2$-norm on unweighted Korobov spaces (Sobolev spaces with mixed smoothness) as well as classical smoothness classes such as Hölder classes suffers from the curse of dimensionality. We show that the problem is tractable for those classes if they are intersected with the Wiener algebra of functions with summable Fourier coefficients. In fact, this is a relatively simple implication of powerful results from the theory of compressed sensing. Tractability is achieved by the use of non-linear algorithms, while linear algorithms cannot do the job.

Theorems & Definitions (24)

• Theorem 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 8
• Lemma 9: Compare RW
• proof
• Proposition 10