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

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 <i>t</i>-sparse polynomial <i>f</i> from a sequence of values, where some of the values are wrong, spoiled by either random or misleading errors. Our algorithm requires bounds <i>T</i> ≥ <i>t</i> and <i>E</i> ≥ <i>e</i>, where <i>e</i> is the number of evaluation errors. It interpolates <i>f</i>(ω^<i>i</i>) for <i>i</i> = 1,..., 2<i>T</i>(<i>E</i> + 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 <i>e</i> ≤ <i>E</i> elements are erroneous. We show that there exist sequences of < 2<i>t</i>(2<i>e</i> + 1) elements, such that two distinct generators of length <i>t</i> satisfy the linear recurrence up to <i>e</i> faults, at least if the field has characteristic ≠ 2. Uniqueness can be proven (for any field characteristic) for length ≥ 2<i>t</i>(2<i>e</i> + 1) of the sequence with <i>e</i> errors. Finally, we present the Majority Rule Berlekamp/Massey algorithm, which can recover the unique minimal linear generator of degree <i>t</i> when given bounds <i>T</i> ≥ <i>t</i> and <i>E</i> ≥ <i>e</i> and the initial sequence segment of 2<i>T</i>(2<i>E</i> + 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.