Sparse polynomial interpolation: compressed sensing, super resolution, or Prony?

Cédric Josz 1 Jean Bernard Lasserre 1 Bernard Mourrain 2
1 LAAS-MAC - Équipe Méthodes et Algorithmes en Commande
LAAS - Laboratoire d'analyse et d'architecture des systèmes [Toulouse]
2 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
CRISAM - Inria Sophia Antipolis - Méditerranée , UoA - University of Athens
Abstract : We show that the sparse polynomial interpolation problem reduces to a discrete super-resolution problem on the n-dimensional torus. Therefore the semidefinite programming approach initiated by Candès & Fernandez-Granda [7] in the univariate case (and later extended to the multi-variate setting) can be applied. In particular, exact recovery is guaranteed provided that a geometric spacing condition on the " supports " holds and the number of evaluations are sufficiently many (but not many). It also turns out that the (compressed sensing) LP-formulation of ℓ 1-norm minimization is also guaranteed to provide exact recovery provided that the evaluations are made in a certain manner and even though the Restricted Isometry Property for exact recovery is not satisfied. (A naive compressed sensing LP-approach does not offer such a guarantee.) Finally we also describe the algebraic Prony method for sparse interpolation, which also recovers the exact decomposition but from less point evaluations and with no geometric spacing condition. We provide two sets of numerical experiments , one in which the super-resolution technique and Prony's method seem to cope equally well with noise, and another in which the super-resolution technique seems to cope with noise better than Prony's method, at the cost of an extra computational burden (i.e. a semidefinite optimization).
Type de document :
Pré-publication, Document de travail
Rapport LAAS n° 17279. 2017
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https://hal.archives-ouvertes.fr/hal-01575325
Contributeur : Jean Bernard Lasserre <>
Soumis le : vendredi 18 août 2017 - 19:35:25
Dernière modification le : samedi 9 septembre 2017 - 01:04:41

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  • HAL Id : hal-01575325, version 1
  • ARXIV : 1708.06187

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Cédric Josz, Jean Bernard Lasserre, Bernard Mourrain. Sparse polynomial interpolation: compressed sensing, super resolution, or Prony?. Rapport LAAS n° 17279. 2017. <hal-01575325>

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