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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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References

  1. Andris Ambainis, Andrew M Childs, Ben W Reichardt, Robert Špalek, and Shengyu Zhang. Any AND-OR formula of size N can be evaluated in time N^1/2+o(1) on a quantum computer. SIAM Journal on Computing, 39(6):2513-2530, 2010. Google Scholar
  2. 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. URL: http://dx.doi.org/10.1109/SFCS.1986.15.
  3. Richard Beigel, Nick Reingold, and Daniel A. Spielman. PP Is Closed under Intersection. J. Comput. Syst. Sci., 50(2):191-202, 1995. URL: http://dx.doi.org/10.1006/jcss.1995.1017.
  4. Arnab Bhattacharyya, Suprovat Ghoshal, and Rishi Saket. Hardness of Learning Noisy Halfspaces using Polynomial Thresholds. In Conference On Learning Theory, COLT 2018, Stockholm, Sweden, 6-9 July 2018., pages 876-917, 2018. URL: http://proceedings.mlr.press/v75/bhattacharyya18a.html.
  5. Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan. On the Power of Statistical Zero Knowledge. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 708-719, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.71.
  6. Mark Bun and Justin Thaler. Improved Bounds on the Sign-Rank of AC⁰. In 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, pages 37:1-37:14, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.37.
  7. Mark Bun and Justin Thaler. The Large-Error Approximate Degree of AC⁰. Electronic Colloquium on Computational Complexity (ECCC), 25:143, 2018. URL: https://eccc.weizmann.ac.il/report/2018/143.
  8. Arkadev Chattopadhyay and Nikhil S. Mande. A Short List of Equalities Induces Large Sign Rank. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 47-58, 2018. URL: http://dx.doi.org/10.1109/FOCS.2018.00014.
  9. Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. New results for learning noisy parities and halfspaces. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 563-574. IEEE, 2006. Google Scholar
  10. Jürgen Forster. A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci., 65(4):612-625, 2002. URL: http://dx.doi.org/10.1016/S0022-0000(02)00019-3.
  11. Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for P^NP. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 12:1-12:16, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2017.12.
  12. Mika Göös, Toniann Pitassi, and Thomas Watson. Query-to-Communication Lifting for BPP. In 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 132-143, 2017. URL: http://dx.doi.org/10.1109/FOCS.2017.21.
  13. Mika Göös, Toniann Pitassi, and Thomas Watson. The Landscape of Communication Complexity Classes. Computational Complexity, 27(2):245-304, 2018. URL: http://dx.doi.org/10.1007/s00037-018-0166-6.
  14. Michael Kearns. Efficient noise-tolerant learning from statistical queries. Journal of the ACM (JACM), 45(6):983-1006, 1998. Google Scholar
  15. Subhash Khot and Rishi Saket. On the hardness of learning intersections of two halfspaces. Journal of Computer and System Sciences, 77(1):129-141, 2011. Google Scholar
  16. Adam R. Klivans, Ryan O'Donnell, and Rocco A. Servedio. Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci., 68(4):808-840, 2004. URL: http://dx.doi.org/10.1016/j.jcss.2003.11.002.
  17. Adam R Klivans and Rocco A Servedio. Learning DNF in time 2^O (n^1/3). Journal of Computer and System Sciences, 68(2):303-318, 2004. Google Scholar
  18. Adam R Klivans and Alexander A Sherstov. Cryptographic hardness for learning intersections of halfspaces. Journal of Computer and System Sciences, 75(1):2-12, 2009. Google Scholar
  19. Marvin Minsky and Seymour Papert. Perceptrons. MIT Press, 1969. Google Scholar
  20. Ryan O'Donnell and Rocco A Servedio. New degree bounds for polynomial threshold functions. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 325-334. ACM, 2003. Google Scholar
  21. Ramamohan Paturi and Janos Simon. Probabilistic Communication Complexity. J. Comput. Syst. Sci., 33(1):106-123, 1986. URL: http://dx.doi.org/10.1016/0022-0000(86)90046-2.
  22. Alexander A. Razborov and Alexander A. Sherstov. The Sign-Rank of AC⁰. SIAM J. Comput., 39(5):1833-1855, 2010. URL: http://dx.doi.org/10.1137/080744037.
  23. Alexander A. Sherstov. The Pattern Matrix Method. SIAM J. Comput., 40(6):1969-2000, 2011. URL: http://dx.doi.org/10.1137/080733644.
  24. Alexander A Sherstov. The unbounded-error communication complexity of symmetric functions. Combinatorica, 31(5):583-614, 2011. Google Scholar
  25. Alexander A. Sherstov. Optimal bounds for sign-representing the intersection of two halfspaces by polynomials. Combinatorica, 33(1):73-96, 2013. URL: http://dx.doi.org/10.1007/s00493-013-2759-7.
  26. Alexander A. Sherstov. The Intersection of Two Halfspaces Has High Threshold Degree. SIAM J. Comput., 42(6):2329-2374, 2013. URL: http://dx.doi.org/10.1137/100785260.
  27. Alexander A Sherstov. Breaking the Minsky-Papert Barrier for Constant-Depth Circuits. SIAM Journal on Computing, 47(5):1809-1857, 2018. Google Scholar
  28. Alexander A. Sherstov and Pei Wu. Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC⁰. CoRR, abs/1901.00988, 2019. To appear in STOC 2019. URL: http://arxiv.org/abs/1901.00988.
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