Sparse polynomial interpolation and Berlekamp/Massey algorithms that correct outlier errors in input values

• Authors: Matthew T. Comer;Erich L. Kaltofen;Clément Pernet
• Venue: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
• Year: 2012

We propose algorithms performing sparse interpolation with errors, based on Prony's--Ben-Or's & Tiwari's algorithm, using a Berlekamp/Massey algorithm with early termination. First, we present an algorithm that can recover a t-sparse polynomial f from a sequence of values, where some of the values are wrong, spoiled by either random or misleading errors. Our algorithm requires bounds T ≥ t and E ≥ e, where e is the number of evaluation errors. It interpolates f(ωi) for i = 1,..., 2T(E + 1), where ω is a field element at which each non-zero term evaluates distinctly. We also investigate the problem of recovering the minimal linear generator from a sequence of field elements that are linearly generated, but where again e ≤ E elements are erroneous. We show that there exist sequences of t(2e + 1) elements, such that two distinct generators of length t satisfy the linear recurrence up to e faults, at least if the field has characteristic ≠ 2. Uniqueness can be proven (for any field characteristic) for length ≥ 2t(2e + 1) of the sequence with e errors. Finally, we present the Majority Rule Berlekamp/Massey algorithm, which can recover the unique minimal linear generator of degree t when given bounds T ≥ t and E ≥ e and the initial sequence segment of 2T(2E + 1) elements. Our algorithm also corrects the sequence segment. The Majority Rule algorithm yields a unique sparse interpolant for the first problem. The algorithms are applied to sparse interpolation algorithms with numeric noise, into which we now can bring outlier errors in the values.