Sampling numbers of smoothness classes via ℓ1-minimization

Kurzfassung/Abstract

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 . 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 can be upper bounded by best n-term trigonometric widths in . We describe a recovery procedure from m function values based on -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 (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 on the d-torus with a logarithmically better rate of convergence than any linear method can achieve when and d is large. This effect is not present for isotropic Sobolev spaces.