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


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Article Info

DATE

2019-03-15

AUTHORS

Cédric Josz, Jean Bernard Lasserre, Bernard Mourrain

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 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 ℓ1-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). More... »

PAGES

1-37

References to SciGraph publications

  • 2015-10. Exact Support Recovery for Sparse Spikes Deconvolution in FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
  • 2014-08. Optimality conditions and finite convergence of Lasserre’s hierarchy in MATHEMATICAL PROGRAMMING
  • 2017-06. Prony’s method in several variables in NUMERISCHE MATHEMATIK
  • 1979. Probabilistic algorithms for sparse polynomials in SYMBOLIC AND ALGEBRAIC COMPUTATION
  • 1989. Improved sparse multivariate polynomial interpolation algorithms in SYMBOLIC AND ALGEBRAIC COMPUTATION
  • 2013-12. Super-Resolution from Noisy Data in JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS
  • 2018-12. Polynomial–Exponential Decomposition From Moments in FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
  • 2009-07. A generalized flat extension theorem for moment matrices in ARCHIV DER MATHEMATIK
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    http://scigraph.springernature.com/pub.10.1007/s10444-019-09672-2

    DOI

    http://dx.doi.org/10.1007/s10444-019-09672-2

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    https://app.dimensions.ai/details/publication/pub.1112775377


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