Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (non-linear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In relevant cases such as mixed and isotropic weighted Wiener classes or Sobolev spaces with mixed smoothness, sampling numbers in $L^2$ can be upper bounded by best $n$-term trigonometric widths in $L^\infty$. We describe a recovery procedure based on $\ell^1$-minimization (basis pursuit denoising) using only $m$ function values. With this method, a significant gain in the rate of convergence compared to recently developed linear recovery methods is achieved. In this deterministic worst-case setting we see an additional speed-up of $n^{-1/2}$ compared to linear methods in case of weighted Wiener spaces. For their quasi-Banach counterparts even arbitrary polynomial speed-up is possible. Surprisingly, our approach allows to recover mixed smoothness Sobolev functions belonging to $S^r_pW(\mathbb{T}^d)$ on the $d$-torus with a logarithmically better rate of convergence than any linear method can achieve when $1 < p < 2$ and $d$ is large. This effect is not present for isotropic Sobolev spaces.

翻译：利用压缩感知领域近年发展的技术，我们给出了（拟）Banach光滑空间在L²中一般（非线性）采样数的新上界。在混合各向同性加权Wiener类或混合光滑Sobolev空间等相关情形中，L²采样数可由L∞中的最佳n项三角宽度给出上界。我们描述了一种基于ℓ¹最小化（基追踪去噪）的重建过程，仅需使用m个函数值。通过该方法，相较于近期发展的线性重建方法，收敛速度获得显著提升。在该确定性最坏情况框架下，对于加权Wiener空间，我们观察到相较于线性方法额外提升n^{-1/2}的速度；对于其拟Banach版本，甚至可实现任意多项式级加速。令人惊讶的是，当1 < p < 2且维数d较大时，我们的方法能够恢复d维环面T^d上属于S^r_pW(T^d)的混合光滑Sobolev函数，其收敛速度比任何线性方法在对数意义上更优。该效应在各项同性Sobolev空间中并不存在。