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

TL;DR

This paper demonstrates that sparse polynomial interpolation can be formulated as a super-resolution problem on the torus, allowing for exact recovery using semidefinite programming and algebraic methods, even with fewer evaluations and under certain conditions.

Contribution

It extends super-resolution techniques to multivariate sparse polynomial interpolation and compares their effectiveness with algebraic Prony's method, providing new guarantees for exact recovery.

Findings

Super-resolution approach guarantees exact recovery under geometric spacing conditions.

Prony's method recovers the exact decomposition with fewer evaluations and no spacing condition.

Super-resolution copes better with noise at higher computational cost.

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} theevaluations 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).