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Improved Bounds on the Sign-Rank of AC^0

Authors Mark Bun, Justin Thaler

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  • Sign-rank
  • circuit complexity
  • communication complexity
  • constant-depth circuits

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Abstract

The sign-rank of a matrix A with entries in {-1, +1} is the least rank of a real matrix B with A_{ij}*B_{ij} > 0 for all i, j. Razborov and Sherstov (2008) gave the first exponential lower bounds on the sign-rank of a function in AC^0, answering an old question of Babai, Frankl, and Simon (1986). Specifically, they exhibited a matrix A = [F(x,y)]_{x,y} for a specific function F:{-1,1}^n*{-1,1}^n -> {-1,1} in AC^0, such that A has sign-rank exp(Omega(n^{1/3}).

We prove a generalization of Razborov and Sherstov’s result, yielding exponential sign-rank lower bounds for a non-trivial class of functions (that includes the function used by Razborov and Sherstov). As a corollary of our general result, we improve Razborov and Sherstov's lower bound on the sign-rank of AC^0 from exp(Omega(n^{1/3})) to exp(~Omega(n^{2/5})). We also describe several applications to communication complexity, learning theory, and circuit complexity.

Cite As Get BibTex

Mark Bun and Justin Thaler. Improved Bounds on the Sign-Rank of AC^0. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.ICALP.2016.37

Author Details

Mark Bun
Justin Thaler

References

  1. Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51(4):595-605, 2004. Google Scholar
  2. Noga Alon, Peter Frankl, and Vojtech Rödl. Geometrical realization of set systems and probabilistic communication complexity. In 26th Annual Symposium on Foundations of Computer Science, Portland, Oregon, USA, 21-23 October 1985, pages 277-280, 1985. Google Scholar
  3. Noga Alon, Shay Moran, and Amir Yehudayoff. Sign rank, VC dimension and spectral gaps. Electronic Colloquium on Computational Complexity (ECCC), 21:135, 2014. Google Scholar
  4. László Babai, Peter Frankl, and Janos Simon. Complexity classes in communication complexity theory (preliminary version). In 27th Annual Symposium on Foundations of Computer Science, Toronto, Canada, 27-29 October 1986, pages 337-347, 1986. Google Scholar
  5. Mark Bun and Justin Thaler. Dual lower bounds for approximate degree and Markov-Bernstein inequalities. In ICALP (1), pages 303-314, 2013. Google Scholar
  6. Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. CoRR, abs/1503.07261, 2015. URL: http://arxiv.org/abs/1503.07261.
  7. Mark Bun and Justin Thaler. Hardness amplification and the approximate degree of constant-depth circuits. In Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part I, pages 268-280, 2015. Google Scholar
  8. Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci., 65(4):612-625, 2002. Google Scholar
  9. Jürgen Forster, Matthias Krause, Satyanarayana V. Lokam, Rustam Mubarakzjanov, Niels Schmitt, and Hans-Ulrich Simon. Relations between communication complexity, linear arrangements, and computational complexity. In FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science, 21st Conference, Bangalore, India, December 13-15, 2001, Proceedings, pages 171-182, 2001. Google Scholar
  10. Jürgen Forster and Hans-Ulrich Simon. On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes uniform distribution. In Algorithmic Learning Theory, 13th International Conference, ALT 2002, Lübeck, Germany, November 24-26, 2002, Proceedings, pages 128-138, 2002. Google Scholar
  11. Mika Göös, Toniann Pitassi, and Thomas Watson. The landscape of communication complexity classes. Electronic Colloquium on Computational Complexity (ECCC), 22:49, 2015. URL: http://eccc.hpi-web.de/report/2015/049.
  12. Lisa Hellerstein and Rocco A. Servedio. On PAC learning algorithms for rich boolean function classes. Theor. Comput. Sci., 384(1):66-76, 2007. Google Scholar
  13. Adam R. Klivans and Rocco A. Servedio. Learning DNF in time 2^õ(n^1/3). J. Comput. Syst. Sci., 68(2):303-318, 2004. Google Scholar
  14. Nati Linial, Shahar Mendelson, Gideon Schechtman, and Adi Shraibman. Complexity measures of sign matrices. Combinatorica, 27(4):439-463, 2007. Google Scholar
  15. Marvin Minsky and Seymour Papert. Perceptrons - an introduction to computational geometry. MIT Press, 1969. Google Scholar
  16. Ramamohan Paturi and Janos Simon. Probabilistic communication complexity. J. Comput. Syst. Sci., 33(1):106-123, 1986. Google Scholar
  17. Alexander A. Razborov and Alexander A. Sherstov. The sign-rank of AC⁰. SIAM J. Comput., 39(5):1833-1855, 2010. Google Scholar
  18. A. A. Sherstov. The power of asymmetry in constant-depth circuits. In Foundations of Computer Science, 2015. Google Scholar
  19. Alexander A. Sherstov. Communication lower bounds using dual polynomials. Bulletin of the EATCS, 95:59-93, 2008. Google Scholar
  20. Alexander A. Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011. Google Scholar
  21. Alexander A. Sherstov. Approximating the AND-OR tree. Theory of Computing, 9(20):653-663, 2013. Google Scholar
  22. Alexander A. Sherstov. The intersection of two halfspaces has high threshold degree. SIAM J. Comput., 42(6):2329-2374, 2013. Google Scholar
  23. Alexander A. Sherstov. Breaking the Minsky-Papert barrier for constant-depth circuits. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 223-232, 2014. Google Scholar
  24. Robert Spalek. A dual polynomial for OR. CoRR, abs/0803.4516, 2008. URL: http://arxiv.org/abs/0803.4516.
  25. Justin Thaler. Lower bounds for the approximate degree of block-composed functions. Electronic Colloquium on Computational Complexity (ECCC), 21:150, 2014. URL: http://eccc.hpi-web.de/report/2014/150.
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