Stability results for random sampling of sparse trigonometric polynomials

TL;DR

This paper demonstrates the stability of reconstructing sparse trigonometric polynomials from noisy samples using Basis Pursuit and Orthogonal Matching Pursuit, extending results to approximately sparse signals with numerical validation.

Contribution

It provides the first stability analysis for BP and OMP in reconstructing sparse and approximately sparse trigonometric polynomials from noisy data.

Findings

BP reconstruction is stable under sample noise.

Partial stability results are shown for OMP.

Numerical experiments confirm theoretical predictions.

Abstract

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.