Sparse polynomial interpolation: sparse recovery, 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 , National and Kapodistrian 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 Cand\`es \& Fernandez-Granda \cite{candes_towards_2014} in the univariate case can be applied. We extend their result to the multivariate case, i.e., we show that 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 sparse recovery LP-formulation of $\ell_1$-norm minimization is also guaranteed to provide exact recovery {\it 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 sparse recovery 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).
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Article dans une revue
Advances in Computational Mathematics, Springer Verlag, 2019, 〈10.1007/s10444-019-09672-2〉
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https://hal.archives-ouvertes.fr/hal-01575325
Contributeur : Jean Bernard Lasserre <>
Soumis le : vendredi 23 novembre 2018 - 13:19:40
Dernière modification le : lundi 18 mars 2019 - 10:51:03

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Cédric Josz, Jean-Bernard Lasserre, Bernard Mourrain. Sparse polynomial interpolation: sparse recovery, super resolution, or Prony?. Advances in Computational Mathematics, Springer Verlag, 2019, 〈10.1007/s10444-019-09672-2〉. 〈hal-01575325v2〉

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