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Sign-Rank Can Increase Under Intersection

Authors Mark Bun, Nikhil S. Mande, Justin Thaler

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LIPIcs.ICALP.2019.30.pdf
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Author Details

Mark Bun
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
  • Boston University, MA, USA
Nikhil S. Mande
  • Georgetown University, Washington, DC, USA
Justin Thaler
  • Georgetown University, Washington, DC, USA

Acknowledgements

Nikhil Mande and Justin Thaler were supported by NSF Grant CCF-1845125.

Cite AsGet BibTex

Mark Bun, Nikhil S. Mande, and Justin Thaler. Sign-Rank Can Increase Under Intersection. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 30:1-30:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.30

Abstract

The communication class UPP^{cc} is a communication analog of the Turing Machine complexity class PP. It is characterized by a matrix-analytic complexity measure called sign-rank (also called dimension complexity), and is essentially the most powerful communication class against which we know how to prove lower bounds. For a communication problem f, let f wedge f denote the function that evaluates f on two disjoint inputs and outputs the AND of the results. We exhibit a communication problem f with UPP^{cc}(f)= O(log n), and UPP^{cc}(f wedge f) = Theta(log^2 n). This is the first result showing that UPP communication complexity can increase by more than a constant factor under intersection. We view this as a first step toward showing that UPP^{cc}, the class of problems with polylogarithmic-cost UPP communication protocols, is not closed under intersection. Our result shows that the function class consisting of intersections of two majorities on n bits has dimension complexity n^{Omega(log n)}. This matches an upper bound of (Klivans, O'Donnell, and Servedio, FOCS 2002), who used it to give a quasipolynomial time algorithm for PAC learning intersections of polylogarithmically many majorities. Hence, fundamentally new techniques will be needed to learn this class of functions in polynomial time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Theory of computation → Boolean function learning
Keywords
  • Sign rank
  • dimension complexity
  • communication complexity
  • learning theory

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