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

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.