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The log-approximate-rank conjecture is false

Published: 23 June 2019 Publication History

Abstract

We construct a simple and total XOR function F on 2n variables that has only O(√n) spectral norm, O(n2) approximate rank and O(n2.5) approximate nonnegative rank. We show it has polynomially large randomized bounded-error communication complexity of Ω(√n). This yields the first exponential gap between the logarithm of the approximate rank and randomized communication complexity for total functions. Thus F witnesses a refutation of the Log-Approximate-Rank Conjecture (LARC) which was posed by Lee and Shraibman as a very natural analogue for randomized communication of the still unresolved Log-Rank Conjecture for deterministic communication. The best known previous gap for any total function between the two measures is a recent 4th-power separation by G'o'os, Jayram, Pitassi and Watson.
Additionally, our function F refutes Grolmusz’s Conjecture and a variant of the Log-Approximate-Nonnegative-Rank Conjecture, suggested recently by Kol, Moran, Shpilka and Yehudayoff, both of which are implied by the LARC. The complement of F has exponentially large approximate nonnegative rank. This answers a question of Lee and Kol et al., showing that approximate nonnegative rank can be exponentially larger than approximate rank. The function F also falsifies a conjecture about parity measures of Boolean functions made by Tsang, Wong, Xie and Zhang. The latter conjecture implied the Log-Rank Conjecture for XOR functions.
We are pleased to note that shortly after we published our results two independent groups of researchers, Anshu, Boddu and Touchette, and Sinha and de Wolf, used our function F to prove that the Quantum-Log-Rank Conjecture is also false by showing that F has Ω(n1/6) quantum communication complexity.

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cover image ACM Conferences
STOC 2019: Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing
June 2019
1258 pages
ISBN:9781450367059
DOI:10.1145/3313276
© 2019 Association for Computing Machinery. ACM acknowledges that this contribution was authored or co-authored by an employee, contractor or affiliate of a national government. As such, the Government retains a nonexclusive, royalty-free right to publish or reproduce this article, or to allow others to do so, for Government purposes only.

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Published: 23 June 2019

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Author Tags

  1. approximate nonnegative rank
  2. log rank conjecture
  3. parity decision trees
  4. randomized communication complexity
  5. spectral norm

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