Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1401–1437 | Cite as

Sparse polynomial interpolation: sparse recovery, super-resolution, or Prony?

  • Cédric Josz
  • Jean Bernard LasserreEmail author
  • Bernard Mourrain


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 and Fernandez-Granda (Commun. Pure Appl. Math. 67(6) 906–956, 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 evaluations are sufficiently many (but not many). It also turns out that the sparse recovery LP-formulation of 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 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).


Linear programming Prony’s method Semidefinite programming super-resolution 

Mathematics Subject Classification (2010)

90-08 90C22 90C25 65K05 


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The research of the second author was funded by the European Research Council (ERC) under the European’s Union Horizon 2020 research and innovation program (grant agreement 666981 TAMING).

Funding information

The work of the first two authors was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement 666981 TAMING).


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Authors and Affiliations

  1. 1.LAAS-CNRSToulouse Cédex 4France
  2. 2.LAAS-CNRS and Institute of MathematicsToulouse Cédex 4France
  3. 3.Université Côte d’Azur, InriaSophia AntipolisFrance

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