Alexander A. Sherstov

The sign-rank of a real matrix M is the least rank

of a matrix R in which every entry has the same sign as the

corresponding entry of M. We determine the sign-rank of every

matrix of the form M=[ D(|x AND y|) ]_{x,y}, where

D:{0,1,...,n}->{-1,+1} is given and ...
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Mark Bun, Justin Thaler

The sign-rank of a matrix $A$ with entries in $\{-1, +1\}$ is the least rank of a real matrix $B$ with $A_{ij} \cdot B_{ij} > 0$ for all $i, j$. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC$^0$, answering an ... more >>>

Hamed Hatami, Kaave Hosseini, Shachar Lovett

Sign-rank and discrepancy are two central notions in communication complexity. The seminal work of Babai, Frankl, and Simon from 1986 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 ...
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Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, Rosie Zhao

The sign-rank of a matrix $A$ with $\pm 1$ entries is the smallest rank of a real matrix with the same sign pattern as $A$. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is ... more >>>

Hamed Hatami, Kaave Hosseini, Xiang Meng

We introduce a new topological argument based on the Borsuk-Ulam theorem to prove a lower bound on sign-rank.

This result implies the strongest possible separation between randomized and unbounded-error communication complexity. More precisely, we show that for a particular range of parameters, the randomized communication complexity of ... more >>>