Halfspace Matrices

Abstract

A halfspace matrix is a Boolean matrix A with rows indexed by linear threshold functions f , columns indexed by inputs x isin {-1,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> , and the entries given by A <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">,x</sub> = f (x). We demonstrate the potential of halfspace matrices as tools to answer nontrivial open questions. (1) (Communication complexity) We exhibit a Boolean function f with discrepancy Omega(1/n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) under every product distribution but O(radicn /2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> /4}) under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan. (2) (Complexity of sign matrices) We construct a matrix A isin {-1,1} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">NtimesN</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">logN</sup> with dimension complexity logN but margin complexity Omega(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/4</sup> /radic{log N}). This gap is an exponential improvement over previous work. As an application to circuit complexity, we prove an Omega(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/4</sup> /(dradicn)) circuit lower bound for computing halfspaces by a majority of an arbitrary set of d gates. This complements a result of Goldmann, Hastad, and Razborov. In addition, we prove new results on the complexity measures of sign matrices, complementing recent work by Linial et al.(3) (Learning theory) We give a short and simple proof that the statistical-query (SQ) dimension of halfspaces in n dimensions is less than 2(n+1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> under all distributions (with n+1 being a trivial lower bound). This improves on the n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(1)</sup> estimate from the fundamental paper of Blum et al. Finally, we motivate our learning-theoretic result for the complexity community by showing that SQ dimension estimates for natural classes of Boolean functions can resolve major open problems in complexity theory. Specifically, we show that an exp(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(logn)</sup> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(1)</sup> ) upper bound on the SQ dimension of AC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> would imply an explicit language in PSPACE <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cc</sup> \PH <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">cc</sup> .