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Sign-Rank Can Increase Under Intersection

Authors Mark Bun, Nikhil S. Mande, Justin Thaler

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

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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.

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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

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.

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