Sign-Rank vs. Discrepancy

Sign-rank and discrepancy are two central notions in communication complexity. The seminal paper by Babai, Frankl, and Simon (FOCS'86) initiated an active line of research that investigates the gap between these two notions. In this article, we establish the strongest possible separation by constructing a boolean matrix whose sign-rank is only 3, and yet its discrepancy is 2-Omega(n). We note that every matrix of sign-rank 2 has discrepancy n-o(1). In connection with learning theory, our result implies the existence of Boolean matrices whose entries are represented by points and half-spaces in dimension 3, and yet, the normalized margin of any such representation (angle between the half-spaces and the unit vectors representing the points), even in higher dimensions, is very small. In the context of communication complexity, our result in particular implies that there are boolean functions with O(1) unbounded-error randomized communication complexity while having Omega(n) weakly unbounded-error randomized communication complexity.