Sparse Polynomial Interpolation by Variable Shift in the Presence of Noise and Outliers in the Evaluations

Abstract

We compute approximate sparse polynomial models of the form $\widetilde{f}(x) = \sum _{j=1}^t \widetilde{c}_j (x - \widetilde{s})^{e_j}$ to a function $f(x)$, of which an approximation of the evaluation $f(\zeta )$ at any complex argument value $\zeta$ can be obtained. We assume that several of the returned function evaluations $f(\zeta )$ are perturbed not just by approximation/noise errors but also by highly perturbed outliers. None of the $\widetilde{c}_j$, $\widetilde{s}$, $e_j$ and the location of the outliers are known beforehand. We use a numerical version of an exact algorithm by [4] together with a numerical version of the Reed–Solomon error correcting coding algorithm. We also compare with a simpler approach based on root finding of derivatives, while restricted to characteristic $0$. In this preliminary report, we discuss how some of the problems of numerical instability and ill-conditioning in the arising optimization problems can be overcome. By way of experiments, we show that our techniques can recover approximate sparse shifted polynomial models, provided that there are few terms $t$, few outliers and that the sparse shift is relatively small.

This material is based on work supported in part by the National Science Foundation under Grants CCF-0830347 and CCF-1115772.