Hardness Amplification for Non-Commutative Arithmetic Circuits
We show that proving mildly super-linear lower bounds on non-commutative arithmetic circuits implies exponential lower bounds on non-commutative circuits. That is, non-commutative circuit complexity is a threshold phenomenon: an apparently weak lower bound actually suffices to show the strongest lower bounds we could desire.
This is part of a recent line of inquiry into why arithmetic circuit complexity, despite being a heavily restricted version of Boolean complexity, still cannot prove super-linear lower bounds on general devices. One can view our work as positive news (it suffices to prove weak lower bounds to get strong ones) or negative news (it is as hard to prove weak lower bounds as it is to prove strong ones). We leave it to the reader to determine their own level of optimism.
arithmetic circuits
hardness amplification
circuit lower bounds
non-commutative computation
Theory of computation~Algebraic complexity theory
12:1-12:16
Regular Paper
Marco L.
Carmosino
Marco L. Carmosino
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
Russell
Impagliazzo
Russell Impagliazzo
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
Shachar
Lovett
Shachar Lovett
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by NSF CAREER award 1350481 and CCF award 1614023
Ivan
Mihajlin
Ivan Mihajlin
Department of Computer Science, University of California San Diego, La Jolla, CA, USA
Supported by the Simons Foundation
10.4230/LIPIcs.CCC.2018.12
Manindra Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2008, October 25-28, 2008, Philadelphia, PA, USA, pages 67-75. IEEE Computer Society, 2008.
Vikraman Arvind, Pushkar S. Joglekar, Partha Mukhopadhyay, and S. Raja. Randomized polynomial time identity testing for noncommutative circuits. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 831-841. ACM, 2017. URL: http://dx.doi.org/10.1145/3055399.3055442.
http://dx.doi.org/10.1145/3055399.3055442
Vikraman Arvind, Pushkar S. Joglekar, and S. Raja. Noncommutative valiant’s classes: Structure and complete problems. TOCT, 9(1):3:1-3:29, 2016. URL: http://dx.doi.org/10.1145/2956230.
http://dx.doi.org/10.1145/2956230
Vikraman Arvind and Srikanth Srinivasan. On the hardness of the noncommutative determinant. In Proceedings of the forty-second ACM symposium on Theory of computing, pages 677-686. ACM, 2010.
Walter Baur and Volker Strassen. The complexity of partial derivatives. Theoretical computer science, 22(3):317-330, 1983.
Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. Electronic Colloquium on Computational Complexity (ECCC), 24:27, 2017.
Michael A. Forbes, Amir Shpilka, and Ben Lee Volk. Succinct hitting sets and barriers to proving algebraic circuits lower bounds. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 653-664. ACM, 2017.
Joshua A. Grochow, Mrinal Kumar, Michael E. Saks, and Shubhangi Saraf. Towards an algebraic natural proofs barrier via polynomial identity testing. Electronic Colloquium on Computational Complexity (ECCC), 24:9, 2017.
Ankit Gupta, Pritish Kamath, Neeraj Kayal, and Ramprasad Saptharishi. Arithmetic circuits: A chasm at depth 3. SIAM J. Comput., 45(3):1064-1079, 2016.
Pavel Hrubes, Avi Wigderson, and Amir Yehudayoff. Relationless completeness and separations. Electronic Colloquium on Computational Complexity (ECCC), 17:40, 2010. URL: http://eccc.hpi-web.de/report/2010/040.
http://eccc.hpi-web.de/report/2010/040
Pavel Hrubeš, Avi Wigderson, and Amir Yehudayoff. Non-commutative circuits and the sum-of-squares problem. Journal of the American Mathematical Society, 24(3):871-898, 2011.
Pascal Koiran. Arithmetic circuits: The chasm at depth four gets wider. Theor. Comput. Sci., 448:56-65, 2012.
François Le Gall. Powers of tensors and fast matrix multiplication. In Proceedings of the 39th international symposium on symbolic and algebraic computation, pages 296-303. ACM, 2014.
Noam Nisan. Lower bounds for non-commutative computation. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 410-418. ACM, 1991.
Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24-35, 1997.
Amir Shpilka, Amir Yehudayoff, et al. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trendsregistered in Theoretical Computer Science, 5(3-4):207-388, 2010.
Leslie G. Valiant. Completeness classes in algebra. In Michael J. Fischer, Richard A. DeMillo, Nancy A. Lynch, Walter A. Burkhard, and Alfred V. Aho, editors, Proceedings of the 11h Annual ACM Symposium on Theory of Computing, April 30 - May 2, 1979, Atlanta, Georgia, USA, pages 249-261. ACM, 1979. URL: http://dx.doi.org/10.1145/800135.804419.
http://dx.doi.org/10.1145/800135.804419
Marco L. Carmosino, Russell Impagliazzo, Shachar Lovett, and Ivan Mihajlin
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode