Stability Results for Random Sampling of Sparse Trigonometric Polynomials

Recently, it has been observed that a sparse trigonometric polynomial, i.e. having only a small number of non-zero coefficients, can be reconstructed exactly from a small number of random samples using Basis Pursuit (BP) or Orthogonal Matching Pursuit (OMP). In the present article it is shown that recovery by a BP variant is stable under perturbation of the samples values by noise. A similar partial result for OMP is provided. For BP in addition, the stability result is extended to (non-sparse) trigonometric polynomials that can be well-approximated by sparse ones. The theoretical findings are illustrated by numerical experiments. Key Words: random sampling, trigonometric polynomials, Orthogonal Matching Pursuit, Basis Pursuit, compressed sensing, stability under noise, fast Fourier transform, non-equispaced fast Fourier transform AMS Subject classification: 94A20, 42A05, 15A52, 90C25

• Type: Journal
• Year: 2008
• Where: TIT
• Authors: Holger Rauhut
• Keywords: Orthogonal Matching Pursuit, Small Number, TIT 2008, Trigonometric Polynomials