Abstract
The communication class UPPcc 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 ∧ 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 UPPcc(f) = O(log n), and UPPcc(f ∧ f) = Θ (log2 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 UPPcc, 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 nOmegaΩ(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.
- Noga Alon, Peter Frankl, and Vojtech Rödl. 1985. Geometrical realization of set systems and probabilistic communication complexity. In Proceedings of the 26th Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 277–280. DOI:DOI:https://doi.org/10.1109/SFCS.1985.30Google Scholar
Digital Library
- Noga Alon, Shay Moran, and Amir Yehudayoff. 2016. Sign rank versus VC dimension. In Proceedings of the 29th Conference on Learning Theory (COLT). JMLR.org, 47–80. Retrieved from http://jmlr.org/proceedings/papers/v49/alon16.html.Google Scholar
- Andris Ambainis, Andrew M. Childs, Ben W. Reichardt, Robert Špalek, and Shengyu Zhang. 2010. Any AND-OR formula of size can be evaluated in time on a quantum computer. SIAM J. Comput. 39, 6 (2010), 2513–2530. DOI:DOI:https://doi.org/10.1137/080712167Google Scholar
Digital Library
- László Babai, Peter Frankl, and Janos Simon. 1986. Complexity classes in communication complexity theory (preliminary version). In Proceedings of the 27th Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 337–347. DOI:DOI:https://doi.org/10.1109/SFCS.1986.15Google Scholar
- Richard Beigel, Nick Reingold, and Daniel A. Spielman. 1995. PP Is closed under intersection. J. Comput. Syst. Sci. 50, 2 (1995), 191–202. DOI:DOI:https://doi.org/10.1006/jcss.1995.1017Google Scholar
Digital Library
- Arnab Bhattacharyya, Suprovat Ghoshal, and Rishi Saket. 2018. Hardness of learning noisy halfspaces using polynomial thresholds. In Proceedings of the Conference On Learning Theory (COLT). PMLR, 876–917. Retrieved from http://proceedings.mlr.press/v75/bhattacharyya18a.html.Google Scholar
- Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan. 2017. On the power of statistical zero knowledge. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 708–719. DOI:DOI:https://doi.org/10.1109/FOCS.2017.71Google Scholar
Cross Ref
- Mark Bun, Nikhil S. Mande, and Justin Thaler. 2019. Sign-rank can increase under intersection. In Proceedings of the 46th International Colloquium on Automata, Languages, and Programming (ICALP). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 30:1–30:14. DOI:DOI:https://doi.org/10.4230/LIPIcs.ICALP.2019.30Google Scholar
- Mark Bun and Justin Thaler. 2016. Improved bounds on the sign-rank of AC. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 37:1–37:14. DOI:DOI:https://doi.org/10.4230/LIPIcs.ICALP.2016.37Google Scholar
- Mark Bun and Justin Thaler. 2019. The large-error approximate degree of AC. In Proceedings of the Conference on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 55:1–55:16. DOI:DOI:https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.55Google Scholar
- Arkadev Chattopadhyay and Nikhil S. Mande. 2018. A short list of equalities induces large sign rank. In Proceedings of the 59th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 47–58. DOI:DOI:https://doi.org/10.1109/FOCS.2018.00014Google Scholar
- Vitaly Feldman, Parikshit Gopalan, Subhash Khot, and Ashok Kumar Ponnuswami. 2006. New results for learning noisy parities and halfspaces. In Proceedings of the 47th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, IEEE Computer Society, 563–574.Google Scholar
Digital Library
- Jürgen Forster. 2002. A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci. 65, 4 (2002), 612–625. DOI:DOI:https://doi.org/10.1016/S0022-0000(02)00019-3Google Scholar
Digital Library
- Jürgen Forster, Matthias Krause, Satyanarayana V. Lokam, Rustam Mubarakzjanov, Niels Schmitt, and Hans Ulrich Simon. 2001. Relations between communication complexity, linear arrangements, and computational complexity. In Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science. Springer, 171–182. DOI:DOI:https://doi.org/10.1007/3-540-45294-X_15Google Scholar
Digital Library
- Jürgen Forster and Hans Ulrich Simon. 2006. On the smallest possible dimension and the largest possible margin of linear arrangements representing given concept classes. Theor. Comput. Sci. 350, 1 (2006), 40–48. DOI:DOI:https://doi.org/10.1016/j.tcs.2005.10.015Google Scholar
Digital Library
- Mika Göös, Pritish Kamath, Toniann Pitassi, and Thomas Watson. 2019. Query-to-communication lifting for P. Comput. Complex. 28, 1 (2019), 113–144. DOI:DOI:https://doi.org/10.1007/s00037-018-0175-5Google Scholar
Digital Library
- Mika Göös, Toniann Pitassi, and Thomas Watson. 2017. Query-to-communication lifting for BPP. In Proceedings of the 58th IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 132–143. DOI:DOI:https://doi.org/10.1109/FOCS.2017.21Google Scholar
Cross Ref
- Mika Göös, Toniann Pitassi, and Thomas Watson. 2018. The landscape of communication complexity classes. Comput. Complex. 27, 2 (2018), 245–304. DOI:DOI:https://doi.org/10.1007/s00037-018-0166-6Google Scholar
Digital Library
- Michael Kearns. 1998. Efficient noise-tolerant learning from statistical queries. J. ACM 45, 6 (1998), 983–1006.Google Scholar
Digital Library
- Subhash Khot and Rishi Saket. 2011. On the hardness of learning intersections of two halfspaces. J. Comput. Syst. Sci. 77, 1 (2011), 129–141.Google Scholar
Digital Library
- Adam R. Klivans, Ryan O’Donnell, and Rocco A. Servedio. 2004. Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci. 68, 4 (2004), 808–840. DOI:DOI:https://doi.org/10.1016/j.jcss.2003.11.002Google Scholar
Digital Library
- Adam R. Klivans and Rocco A. Servedio. 2004. Learning DNF in time . J. Comput. Syst. Sci. 68, 2 (2004), 303–318. DOI:DOI:https://doi.org/10.1016/j.jcss.2003.07.007Google Scholar
Digital Library
- Adam R. Klivans and Alexander A. Sherstov. 2009. Cryptographic hardness for learning intersections of halfspaces. J. Comput. Syst. Sci. 75, 1 (2009), 2–12.Google Scholar
Digital Library
- Nati Linial, Shahar Mendelson, Gideon Schechtman, and Adi Shraibman. 2007. Complexity measures of sign matrices. Combinatorica 27, 4 (2007), 439–463.Google Scholar
Digital Library
- Marvin Minsky and Seymour Papert. 1969. Perceptrons. The MIT Press.Google Scholar
- Ramamohan Paturi and Janos Simon. 1986. Probabilistic communication complexity. J. Comput. Syst. Sci. 33, 1 (1986), 106–123. DOI:DOI:https://doi.org/10.1016/0022-0000(86)90046-2Google Scholar
Digital Library
- Alexander A. Razborov and Alexander A. Sherstov. 2010. The sign-rank of AC. SIAM J. Comput. 39, 5 (2010), 1833–1855. DOI:DOI:https://doi.org/10.1137/080744037Google Scholar
Digital Library
- Alexander A. Sherstov. 2011. The pattern matrix method. SIAM J. Comput. 40, 6 (2011), 1969–2000. DOI:DOI:https://doi.org/10.1137/080733644Google Scholar
Digital Library
- Alexander A. Sherstov. 2011. The unbounded-error communication complexity of symmetric functions. Combinatorica 31, 5 (2011), 583–614. DOI:DOI:https://doi.org/10.1007/s00493-011-2580-0Google Scholar
Digital Library
- Alexander A. Sherstov. 2013. The intersection of two halfspaces has high threshold degree. SIAM J. Comput. 42, 6 (2013), 2329–2374. DOI:DOI:https://doi.org/10.1137/100785260Google Scholar
Cross Ref
- Alexander A. Sherstov. 2013. Optimal bounds for sign-representing the intersection of two halfspaces by polynomials. Combinatorica 33, 1 (2013), 73–96. DOI:DOI:https://doi.org/10.1007/s00493-013-2759-7Google Scholar
Digital Library
- Alexander A. Sherstov. 2018. Breaking the Minsky-Papert barrier for constant-depth circuits. SIAM J. Comput. 47, 5 (2018), 1809–1857. DOI:DOI:https://doi.org/10.1137/15M1015704Google Scholar
Cross Ref
- Alexander A. Sherstov and Pei Wu. 2019. Near-optimal lower bounds on the threshold degree and sign-rank of AC. In Proceedings of the 51st ACM SIGACT Symposium on Theory of Computing (STOC). ACM, 401–412. DOI:DOI:https://doi.org/10.1145/3313276.3316408Google Scholar
Index Terms
Sign-rank Can Increase under Intersection
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