Sparse multivariate function recovery with a small number of evaluations

Abstract

In [Kaltofen and Yang, Proc. ISSAC 2014] we give an algorithm based algebraic error-correcting decoding for multivariate sparse rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors ("outliers"). Our 2014 algorithm can interpolate a sparse multivariate rational function from evaluations where the error rate is 1/q is quite high, say q = 5.For the algorithm with exact arithmetic and exact values at non-erroneous points, one avoids quadratic oversampling by using random evaluation points. Here we give the full probabilistic analysis for this fact, thus providing the missing proof to Theorem 2.1 in Section 2 of our ISSAC 2014 paper. Our argumentation already applies to our original 2007 sparse rational function interpolation algorithm [Kaltofen, Yang and Zhi, Proc. SNC 2007], where we have experimentally observed that for T unknown non-zero coefficients in a sparse candidate ansatz one only needs T + O(1) evaluations rather than O(T 2 ) (cf. Candès and Tao sparse sensing), the latter of which we have proved in 2007. Here we prove that T + O(1) evaluations at random points indeed suffice.

Prony's method in several variables: Symbolic solutions by universal interpolation

“…The closely related problem of reconstruction of sparse polynomials in several variables has also been considered by methods other than Prony's approach. Probabilistic methods with a small expected number of evaluations can be found in (Zippel, 1979), see also (Zippel, 1990) and, more recently, in (Kaltofen and Yang, 2016).…”