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

Authors:
Alexander A. Sherstov
Affiliations:
Microsoft Research, Cambridge, MA, USA
Venue:
Proceedings of the forty-second ACM symposium on Theory of computing
Year:
2010

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Abstract

The threshold degree of a function f:{0,1}n-{-1,+1} is the least degree of a real polynomial p with f=sgn p. 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 2Θ(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.