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) ? sgnp(x). We prove that the intersection of two halfspaces on {0,1} n has threshold degree ?(n), which matches the trivial upper bound and solves an open problem due to Klivans (2002). The best previous lower bound was ?({ie73-1}). Our result shows that the intersection of two halfspaces on {0,1} n only admits a trivial {ie73-2}-time learning algorithm based on sign-representation by polynomials, unlike the advances achieved in PAC learning DNF formulas and read-once Boolean formulas.
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