Sparse Interpolation With Errors in Chebyshev Basis Beyond Redundant-Block Decoding

TL;DR

This paper introduces efficient sparse polynomial interpolation algorithms capable of correcting errors in evaluations, applicable to both standard power and Chebyshev bases, with fewer samples than previous methods.

Contribution

The authors develop new polynomial interpolation algorithms that reduce the number of required samples for error correction in both power and Chebyshev bases, improving upon prior work.

Findings

Fewer samples needed than previous algorithms for error correction.

Able to correct multiple errors in polynomial evaluations.

Applicable to both standard power and Chebyshev polynomial bases.

Abstract

We present sparse interpolation algorithms for recovering a polynomial with $\le B$ terms from $N$ evaluations at distinct values for the variable when $\le E$ of the evaluations can be erroneous. Our algorithms perform exact arithmetic in the field of scalars $\mathsf{K}$ and the terms can be standard powers of the variable or Chebyshev polynomials, in which case the characteristic of $\mathsf{K}$ is $\ne 2$. Our algorithms return a list of valid sparse interpolants for the $N$ support points and run in polynomial-time. For standard power basis our algorithms sample at $N = \lfloor \frac{4}{3} E + 2 \rfloor B$ points, which are fewer points than $N = 2(E+1)B - 1$ given by Kaltofen and Pernet in 2014. For Chebyshev basis our algorithms sample at $N = \lfloor \frac{3}{2} E + 2 \rfloor B$ points, which are also fewer than the number of points required by the algorithm given by Arnold and Kaltofen in 2015, which has $N = 74 \lfloor \frac{E}{13} + 1 \rfloor$ for $B = 3$ and $E \ge 222$. Our method shows how to correct $2$ errors in a block of $4B$ points for standard basis and how to correct $1$ error in a block of $3B$ points for Chebyshev Basis.