Halfspace Matrices

Abstract.

We introduce the notion of a halfspace matrix, which is a sign matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x ∈ {− 1, 1}ⁿ, and the entries given by A _f,x = f(x). We use halfspace matrices to solve the following problems.

In communication complexity, we exhibit a Boolean function f with discrepancy Ω(1/n ⁴) under every product distribution but $O(\sqrt{n}/2^{n/4})$ under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan (1997).

In learning theory, we give a short and simple proof that the statisticalquery (SQ) dimension of halfspaces in n dimensions is less than 2$(n + 1)^2$ under all distributions. This improves on the $n^{O(1)}$ estimate from the fundamental paper of Blum et al. (1998). We show that estimating the SQ dimension of natural classes of Boolean functions can resolve major open problems in complexity theory, such as separating PSPACE $^{cc}$ and PH $^{cc}$.

Finally, we construct a matrix $A \in \{-1, 1\}^{N \times N^{\text{log} N}}$ with dimension complexity logN but margin complexity $\Omega(N^{1/4}/\sqrt{\log N})$. This gap is an exponential improvement over previous work. We prove several other relationships among the complexity measures of sign matrices, omplementing work by Linial et al. (2006, 2007).