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Using techniques developed recently in the field of compressed sensing weprove new upper bounds for general (nonlinear) sampling numbers of(quasi-)Banach smoothness spaces in $L^2$. In particular, we show that inrelevant cases such as mixed and isotropic weighted Wiener classes or Sobolevspaces with mixed smoothness, sampling numbers in $L^2$ can be upper bounded bybest $n$-term trigonometric widths in $L^\infty$. We describe a recoveryprocedure from $m$ function values based on $\ell^1$-minimization (basispursuit denoising). With this method, a significant gain in the rate ofconvergence 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 weightedWiener spaces. For their quasi-Banach counterparts even arbitrary polynomialspeed-up is possible. Surprisingly, our approach allows to recover mixedsmoothness Sobolev functions belonging to $S^r_pW(\mathbb{T}^d)$ on the$d$-torus with a logarithmically better rate of convergence than any linearmethod can achieve when $1 < p < 2$ and $d$ is large. This effect is notpresent for isotropic Sobolev spaces.

Sampling numbers of smoothness classes via $\ell^1$-minimization