ABSTRACT
Algebraic natural proofs were recently introduced by Forbes, Shpilka and Volk (Proc. of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pages 653–664, 2017) and independently by Grochow, Kumar, Saks and Saraf (CoRR, abs/1701.01717, 2017) as an attempt to transfer Razborov and Rudich’s famous barrier result (J. Comput. Syst. Sci., 55(1): 24–35, 1997) for Boolean circuit complexity to algebraic complexity theory. Razborov and Rudich’s barrier result relies on a widely believed assumption, namely, the existence of pseudo-random generators. Unfortunately, there is no known analogous theory of pseudo-randomness in the algebraic setting. Therefore, Forbes et al. use a concept called succinct hitting sets instead. This assumption is related to polynomial identity testing, but it is currently not clear how plausible this assumption is. Forbes et al. are only able to construct succinct hitting sets against rather weak models of arithmetic circuits.
Generalized matrix completion is the following problem: Given a matrix with affine linear forms as entries, find an assignment to the variables in the linear forms such that the rank of the resulting matrix is minimal. We call this rank the completion rank. Computing the completion rank is an NP-hard problem. As our first main result, we prove that it is also NP-hard to determine whether a given matrix can be approximated by matrices of completion rank ≤ b. The minimum quantity b for which this is possible is called border completion rank (similar to the border rank of tensors). Naturally, algebraic natural proofs can only prove lower bounds for such border complexity measures. Furthermore, these border complexity measures play an important role in the geometric complexity program.
Using our hardness result above, we can prove the following barrier: We construct a small family of matrices with affine linear forms as entries and a bound b, such that at least one of these matrices does not have an algebraic natural proof of polynomial size against all matrices of border completion rank b, unless coNP ⊆ ∃ BPP. This is an algebraic barrier result that is based on a well-established and widely believed conjecture. The complexity class ∃ BPP is known to be a subset of the more well known complexity class in the literature. Thus ∃ BPP can be replaced by MA in the statements of all our results. With similar techniques, we can also prove that tensor rank is hard to approximate.
Furthermore, we prove a similar result for the variety of matrices with permanent zero. There are no algebraic polynomial size natural proofs for the variety of matrices with permanent zero, unless P#P ⊆ ∃ BPP. On the other hand, we are able to prove that the geometric complexity theory approach initiated by Mulmuley and Sohoni (SIAM J. Comput. 31(2): 496–526, 2001) yields proofs of polynomial size for this variety, therefore overcoming the natural proofs barrier in this case.
Supplemental Material
- Scott Aaronson and Andrew Drucker. Algebraic natural proofs theory is sought. Blog post at http://www.scottaaronson.com/blog/?p=336, 2008.Google Scholar
- Scott Aaronson and Andrew Drucker. Impagliazzo’s worlds in arithmetic complexity. Talk presented at the Workshop on Complexity and Cryptography: Status of Impagliazzo’s Worlds, Center for Computational Intractability, Princeton, NJ, June 5, 2009. Slides available at http://www.scottaaronson.com/talks/arith.ppt, 2009.Google Scholar
- G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation. Springer, 1999. Google ScholarDigital Library
- Markus Bläser. Fast matrix multiplication. Theory of Computing, Graduate Surveys, 5:1–60, 2013.Google Scholar
- Markus Bläser and Christian Ikenmeyer. Introduction to geometric complexity theory. Lecture notes: https://people.mpiinf.mpg.de/~cikenmey/teaching/ summer17/introtogct/gct.pdf, 2017.Google Scholar
- P. Bürgisser. Completeness and Reduction in Algebraic Complexity Theory. Algorithms and Computation in Mathematics. Springer Berlin Heidelberg, 2000. ISBN 9783540667520.Google Scholar
- Peter Bürgisser and Christian Ikenmeyer. Explicit lower bounds via geometric complexity theory. In Dan Boneh, Tim Roughgarden, and Joan Feigenbaum, editors, Symposium on Theory of Computing Conference, STOC’13, Palo Alto, CA, USA, June 1-4, 2013, pages 141–150. ACM, 2013. ISBN 978-1-4503-2029-0.Google Scholar
- Peter Bürgisser, Michael Clausen, and Mohammad Amin Shokrollahi. Algebraic complexity theory, volume 315 of Grundlehren der mathematischen Wissenschaften. Springer, 1997. ISBN 3-540-60582-7.Google ScholarCross Ref
- Harm Derksen. On the equivalence between low rank matrix completion and tensor rank. CoRR, abs/1406.0080, 2014.Google Scholar
- Klim Efremenko, Ankit Garg, Rafael Mendes de Oliveira, and Avi Wigderson. Barriers for rank methods in arithmetic complexity. CoRR, abs/1710.09502, 2017.Google Scholar
- Michael A. Forbes, Amir Shpilka, and Ben Lee Volk. Succinct hitting sets and barriers to proving algebraic circuits lower bounds. In Hatami et al. { 23 }, pages 653–664. ISBN 978-1-4503-4528-6. Google ScholarDigital Library
- Hervé Fournier, Sylvain Perifel, and Rémi de Verclos. On fixed-polynomial size circuit lower bounds for uniform polynomials in the sense of valiant. In Krishnendu Chatterjee and Jirí Sgall, editors, Mathematical Foundations of Computer Science 2013: 38th International Symposium, MFCS 2013, Klosterneuburg, Austria, August 26-30, 2013. Proceedings, pages 433–444. Springer, 2013. ISBN 978-3-642-40313-2.Google ScholarCross Ref
- William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1997. ISBN 0-521-56144-2; 0-521-56724-6.Google Scholar
- William Fulton and Joe Harris. Representation Theory - A First Course, volume 129 of Graduate Texts in Mathematics. Springer, 1991.Google Scholar
- D.A. Gay. Characters of the Weyl group of SU (n) on zero weight spaces and centralizers of permutation representations. Rocky Mountain J. Math., 6(3):449– 455, 1976. ISSN 0035-7596.Google ScholarCross Ref
- Joshua A. Grochow and Toniann Pitassi. Circuit complexity, proof complexity, and polynomial identity testing. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 110–119. IEEE Computer Society, 2014. ISBN 978-1-4799-6517-5. Google ScholarDigital Library
- Joshua A. Grochow, Ketan D. Mulmuley, and Youming Qiao. Boundaries of VP and VNP. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming, ICALP 2016, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 34:1–34:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. ISBN 978-3-95977-013-2.Google Scholar
- Joshua A. Grochow, Mrinal Kumar, Michael E. Saks, and Shubhangi Saraf. Towards an algebraic natural proofs barrier via polynomial identity testing. CoRR, abs/1701.01717, 2017.Google Scholar
- Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in quasi-NC. In Hatami et al. { 23 }, pages 821–830. ISBN 978-1-4503-4528-6. Google ScholarDigital Library
- Moritz Hardt, Raghu Meka, Prasad Raghavendra, and Benjamin Weitz. Computational limits for matrix completion. In Maria-Florina Balcan, Vitaly Feldman, and Csaba Szepesvári, editors, Proceedings of The 27th Conference on Learning Theory, COLT 2014, Barcelona, Spain, June 13-15, 2014, volume 35 of JMLR Workshop and Conference Proceedings, pages 703–725. JMLR.org, 2014.Google Scholar
- Nicholas J. A. Harvey, David R. Karger, and Sergey Yekhanin. The complexity of matrix completion. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, Miami, Florida, USA, January 22-26, 2006, pages 1103–1111. ACM Press, 2006. ISBN 0-89871-605-5. Google ScholarDigital Library
- Johan Håstad. Tensor rank is NP-complete. J. Algorithms, 11(4):644–654, 1990. Google ScholarDigital Library
- 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, 2017. ACM. ISBN 978-1-4503-4528-6. Google ScholarDigital Library
- Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex., 13(1/2):1– 46, December 2004. ISSN 1016-3328. Google ScholarDigital Library
- Erich Kaltofen. Factorization of polynomials given by straight-line programs. In Randomness and Computation, pages 375–412. JAI Press, 1989.Google Scholar
- Guillaume Malod and Natacha Portier. Characterizing Valiant’s algebraic complexity classes. Journal of complexity, 24(1):16–38, 2008. Google ScholarDigital Library
- K.D. Mulmuley and M. Sohoni. Geometric Complexity Theory. II. Towards explicit obstructions for embeddings among class varieties. SIAM J. Comput., 38 (3):1175–1206, 2008. Google ScholarDigital Library
- Ketan Mulmuley. The GCT program toward the P vs. NP problem. Commun. ACM, 55(6):98–107, 2012. Google ScholarDigital Library
- Ketan Mulmuley and Milind A. Sohoni. Geometric complexity theory I: an approach to the P vs. NP and related problems. SIAM J. Comput., 31(2): 496–526, 2001. Google ScholarDigital Library
- René Peeters. Orthogonal representations over finite fields and the chromatic number of graphs. Combinatorica, 16(3):417–431, 1996.Google ScholarCross Ref
- Alexander A. Razborov and Steven Rudich. Natural proofs. J. Comput. Syst. Sci., 55(1):24–35, 1997. Google ScholarDigital Library
- Bruce E. Sagan. The symmetric group, volume 203 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2001. ISBN 0-387-95067-2. Representations, combinatorial algorithms, and symmetric functions.Google ScholarCross Ref
- Marcus Schaefer and Daniel Stefankovic. The complexity of tensor rank. CoRR, abs/1612.04338, 2016.Google Scholar
- Yarolav Shitov. How hard is the tensor rank? CoRR, abs/1611.01559, 2016.Google Scholar
- Zhao Song, David P. Woodruff, and Peilin Zhong. Relative error tensor low rank approximation. CoRR, abs/1704.08246, 2017.Google Scholar
Index Terms
- Generalized matrix completion and algebraic natural proofs
Recommendations
Natural Proofs
Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995We introduce the notion ofnaturalproof. We argue that the known proofs of lower bounds on the complexity of explicit Boolean functions in nonmonotone models fall within our definition of natural. We show, based on a hardness assumption, that natural ...
Almost-Natural Proofs
FOCS '08: Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer ScienceRazborov and Rudich have shown that so-called ``natural proofs'' are not useful for separating P from NP unless hard pseudorandom number generators do not exist. This famous result is widely regarded as a serious barrier to proving strong lower bounds ...
Natural proofs versus derandomization
STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of ComputingWe study connections between Natural Proofs, derandomization, and the problem of proving "weak" circuit lower bounds such as NEXP ⊄ TC0, which are still wide open. Natural Proofs have three properties: they are constructive (an efficient algorithm A is ...
Comments