Numerical Sparsity Determination and Early Termination

Abstract

Ankur Moitra in his paper at STOC 2015 has given an in-depth analysis of how oversampling improves the conditioning of the arising Prony systems for sparse interpolation and signal recovery from numeric data. Moitra assumes that oversampling is done for a number of samples beyond the actual sparsity of the polynomial/signal. We give an algorithm that can be used to compute the sparsity and estimate the minimal number of samples needed in numerical sparse interpolation. The early termination strategy of polynomial interpolation has been incorporated in the algorithm: by oversampling at a small number of extra sample points we can diagnose that the sparsity has not been reached. Our algorithm still has to make a guess, the number ζ of oversamples, and we show by example that if ζ is guessed too small, premature termination can occur, but our criterion is numerically more accurate than that by Kaltofen, Lee and Yang (Proc. SNC 2011, ACM [12]), but not as efficiently computable. For heuristic justification one has available the multivariate early termination theorem by Kaltofen and Lee (JSC vol. 36(3--4) 2003 [11]) for exact arithmetic, and the numeric Schwartz-Zippel Lemma by Kaltofen, Yang and Zhi (Proc. SNC 2007, ACM [13]). A main contribution here is a modified proof of the Theorem by Kaltofen and Lee that permits starting the sequence at the point (1,...,1), for scalar fields of characteristic ≠ 2 (in characteristic 2 counter-examples are given).