Optimal bounds for sign-representing the intersection of two halfspaces by polynomials

TL;DR

This paper establishes that the threshold degree of the intersection of two halfspaces on {0,1}^n is linear in n, indicating inherent complexity in their polynomial sign-representation and impacting learning algorithms.

Contribution

It proves the first linear lower bound of Omega(n) for the threshold degree of intersecting two halfspaces, resolving a longstanding open problem.

Findings

Threshold degree of intersection is Omega(n).

Implicates trivial learning complexity for such intersections.

Introduces new Fourier and matrix analysis techniques.

Abstract

The threshold degree of a function f:{0,1}^n->{-1,+1} is the least degree of a real polynomial p with f(x)=sgn p(x). We prove that the intersection of two halfspaces on {0,1}^n has threshold degree Omega(n), which matches the trivial upper bound and completely answers a question due to Klivans (2002). The best previous lower bound was Omega(sqrt n). Our result shows that the intersection of two halfspaces on {0,1}^n only admits a trivial 2^{Theta(n)}-time learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC learning DNF formulas and read-once Boolean formulas. The proof introduces a new technique of independent interest, based on Fourier analysis and matrix theory.