Using techniques developed recently in the field of compressed sensing we prove new upper bounds for general (nonlinear) sampling numbers of (quasi-)Banach smoothness spaces in $L^2$. In particular, we show that 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 from $m$ function values based on $\ell^1$-minimization (basis pursuit denoising). 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 $m^{-1/2}$ (up to log factors) 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来发展的技术，我们证明了(拟)巴拿赫光滑空间在$L^2$中一般(非线性)采样数的新上界。特别地，我们表明在相关情形(如混合与各向同性加权Wiener类或混合光滑Sobolev空间)中，$L^2$采样数可由$L^\infty$中的最佳$n$项三角宽度上界控制。我们描述了一种基于$\ell^1$极小化(基追踪去噪)从$m$个函数值进行恢复的过程。通过该方法，与近年来发展的线性恢复方法相比，收敛速率获得了显著提升。在此确定的最坏情形设定中，对于加权Wiener空间，我们看到相较于线性方法额外获得了$m^{-1/2}$(至多对数因子)的加速。而对于对应的拟巴拿赫空间，甚至可实现任意多项式的加速。令人惊讶的是，我们的方法能够以比任何线性方法在对数意义上更优的收敛速率恢复属于$d$维环面上$S^r_pW(\mathbb{T}^d)$的混合光滑Sobolev函数，当$1 < p < 2$且$d$较大时。此效应在各项同性Sobolev空间中并不存在。