Approximate Nonnegative Rank Is Equivalent to the Smooth Rectangle Bound

  • Gillat Kol
  • Shay Moran
  • Amir Shpilka
  • Amir Yehudayoff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)


We consider two known lower bounds on randomized communication complexity: The smooth rectangle bound and the logarithm of the approximate nonnegative rank. Our main result is that they are the same up to a multiplicative constant and a small additive term.

The logarithm of the nonnegative rank is known to be a nearly tight lower bound on the deterministic communication complexity. Our result indicates that proving the analogue for the randomized case, namely that the log approximate nonnegative rank is a nearly tight bound on randomized communication complexity, would imply the tightness of the information cost bound.

Another corollary of our result is the existence of a boolean function with a quasipolynomial gap between its approximate rank and approximate nonnegative rank.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Gillat Kol
    • 1
  • Shay Moran
    • 2
  • Amir Shpilka
    • 3
  • Amir Yehudayoff
    • 4
  1. 1.School of MathematicsInstitute for Advanced StudyPrincetonUSA
  2. 2.Departments of Computer Science and MathematicsTechnionIsrael
  3. 3.Department of Computer ScienceTechnionHaifaIsrael
  4. 4.Department of MathematicsTechnionHaifaIsrael

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