Hardness of Reconstructing Multivariate Polynomials over Finite Fields

We study the polynomial reconstruction problem for low-degree multivariate polynomials over finite field F[2]. In this problem, we are given a set of points x ∈ {0, 1}n and target values f(x) ∈ {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 − ε fraction of the points. Our goal is to find a degree d polynomial that has good agreement with f. We show that it is NP-hard to find a polynomial that agrees with f on more than 1 − 2−d + δ fraction of the points for any , δ > 0. This holds even with the stronger promise that the polynomial that fits the data is in fact linear, whereas the algorithm is allowed to find a polynomial of degree d. Previously the only known hardness of approximation (or even NP-completeness) was for the case when d = 1, which follows from a celebrated result of H˚astad [H˚as01]. In the setting of Computational Learning, our result shows the hardness of non-pro...

• Type: Conference
• Year: 2007
• Where: FOCS
• Authors: Parikshit Gopalan, Subhash Khot, Rishi Saket
• Keywords: FOCS 2007, Polynomial, Polynomial Reconstruction, Polynomial Reconstruction Problem, Theoretical Computer Science