Sign Rank vs Discrepancy

Authors Hamed Hatami, Kaave Hosseini, Shachar Lovett

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Author Details

Hamed Hatami
  • McGill University, Montreal, Canada
Kaave Hosseini
  • Carnegie Mellon University, Pittsburgh, PA, USA
Shachar Lovett
  • University of California San Diego, CA, USA

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Hamed Hatami, Kaave Hosseini, and Shachar Lovett. Sign Rank vs Discrepancy. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 18:1-18:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 is only 3, and yet its discrepancy is 2^{-Ω(n)}. We note that every matrix of sign-rank 2 has discrepancy n^{-O(1)}. Our result in particular implies that there are boolean functions with O(1) unbounded error randomized communication complexity while having Ω(n) weakly unbounded error randomized communication complexity.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Discrepancy
  • sign rank
  • Unbounded-error communication complexity
  • weakly unbounded error communication complexity


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