Theory of Computing
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Title     : Sign-Rank vs. Discrepancy
Authors   : Kaave Hosseini, Hamed Hatami, and Shachar Lovett
Volume    : 18
Number    : 19
Pages     : 1-22
URL       : https://theoryofcomputing.org/articles/v018a019

Abstract
--------

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.

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A conference version of this paper appeared in the Proceedings
of the 35th Computational Complexity Conference, 2020 (CCC'20).