Sampling numbers of smoothness classes via ℓ^1-minimization
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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^∞. We describe a recovery procedure based on ℓ^1-minimization (basis pursuit denoising) using only m function values with m close to n. 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(𝕋^d) on the d-torus with a logarithmically better error decay than any linear method can achieve when 1 < p < 2 and d is large. This effect is not present for isotropic Sobolev spaces.
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