Sampling numbers of smoothness classes via ℓ1-minimization

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 L2. 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 L2 can be upper bounded by best n-term trigonometric widths in L∞. We describe a recovery procedure from m function values based on ℓ1-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 SprW(Td) 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.