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

Abstract

We show that the sparse polynomial interpolation problem reduces to a discrete superresolution problem on the n-dimensional torus. Therefore the semidefinite programming approach initiated by 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 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 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).

The Moment-Sos Hierarchy

“…Here the goal is to recover an unknown (black-box) polynomial p ∈ R[x] t through a few evaluations of p only. In [16] we have shown that this problem is in fact a particular case of Super-Resolution (and even discrete Super-Resolution) on the torus T n ⊂ C n . Indeed let z 0 ∈ T n be fixed, arbitrary.…”

“…Hence one may recover p by solving the Moment-SOS hierarchy (3.13) for which finite convergence usually occurs fast. For more details see [16].…”

The dual approach to non-negative super-resolution: impact on primal reconstruction accuracy

“…Definition 1. (Dual certificate) Consider a solution λ˚of the dual problem (10) or (24). Then a dual certificate is a function of the form…”

“…Lastly, note that the results in Section 2 and Section 3 refer to different optimisation problems: the duals (10) and (24) of problems (7) and (9) respectively. However, the proofs of our perturbation results rely on the property that the dual solution λ forms a dual certificate, the global maximisers of which give the locations of the point sources in the input signal x, with the additional bound on λ from (24) being used in the proof of Theorem 4.…”

“…In this section, we present numerical experiments which verify the bounds given by our main results, Theorem 2, Theorem 3 and Theorem 4. To do this, we take an example of a source and sample configuration and a Gaussian kernel for a given σ and solve the exact penalty formulation (25) of the dual problem (24) using the level method [13], given in Appendix B. We introduce inaccuracies in λ by stopping the algorithm early and show how these perturbations affect the source locations and weights.…”