Exact Reconstruction using Beurling Minimal Extrapolation

Yohann de Castro (1) , Fabrice Gamboa (2, 3)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc.). Generalized minimal extrapolation shares related geometric properties with the basis pursuit approach of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points, can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.

Domaines: Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]; Théorie [stat.TH]

Yohann de Castro, Fabrice Gamboa. Exact Reconstruction using Beurling Minimal Extrapolation. Journal of Mathematical Analysis and Applications, 2012, 395 (1), pp.Pages 336-354. ⟨hal-00678423⟩