The Intersection of Two Halfspaces Has High Threshold Degree

The threshold degree of a Boolean function $f\!\!\,: \{0,1\}^n \rightarrow \{-1, +1\}$ is the least degree of a real polynomial $p$ such that $f(x)\equiv{{\rm sgn}\,p(x)}.$ We construct two halfspaces on $\{0,1\}^n$ whose intersection has threshold degree $\...

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 ...

We consider the complexity of properly learning concept classes, i.e. when the learner must output a hypothesis of the same form as the unknown concept. We present the following new upper and lower bounds on well-known concept classes:*We show that ...