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Efficient Reconstruction of Depth Three Arithmetic Circuits with Top Fan-In Two

Author Gaurav Sinha

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  • Theory of computation → Algebraic complexity theory
Keywords
  • Arithmetic Circuits
  • Circuit Reconstruction

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Abstract

In this paper we develop efficient randomized algorithms to solve the black-box reconstruction problem for polynomials over finite fields, computable by depth three arithmetic circuits with alternating addition/multiplication gates, such that output gate is an addition gate with in-degree two. Such circuits naturally compute polynomials of the form G×(T₁ + T₂), where G,T₁,T₂ are product of affine forms computed at the first layer in the circuit, and polynomials T₁,T₂ have no common factors. Rank of such a circuit is defined to be the dimension of vector space spanned by all affine factors of T₁ and T₂. For any polynomial f computable by such a circuit, rank(f) is defined to be the minimum rank of any such circuit computing it. Our work develops randomized reconstruction algorithms which take as input black-box access to a polynomial f (over finite field 𝔽), computable by such a circuit. Here are the results. 
- [Low rank]: When 5 ≤ rank(f) = O(log³ d), it runs in time (nd^{log³d}log |𝔽|)^{O(1)}, and, with high probability, outputs a depth three circuit computing f, with top addition gate having in-degree ≤ d^{rank(f)}. 
- [High rank]: When rank(f) = Ω(log³ d), it runs in time (ndlog |𝔽|)^{O(1)}, and, with high probability, outputs a depth three circuit computing f, with top addition gate having in-degree two. 
Prior to our work, black-box reconstruction for this circuit class was addressed in [Amir Shpilka, 2007; Karnin and Shpilka, 2009; Sinha, 2016]. Reconstruction algorithm in [Amir Shpilka, 2007] runs in time quasi-polynomial in n,d,|𝔽| and that in [Karnin and Shpilka, 2009] is quasi-polynomial in d,|𝔽|. Algorithm in [Sinha, 2016] works only for polynomials over characteristic zero fields. Thus, ours is the first blackbox reconstruction algorithm for this class of circuits that runs in time polynomial in log |𝔽|. This problem has been mentioned as an open problem in [Ankit Gupta et al., 2012] (STOC 2012). In the high rank case, our algorithm runs in (ndlog|𝔽|)^{O(1)} time, thereby significantly improving the existing algorithms in [Amir Shpilka, 2007; Karnin and Shpilka, 2009].

Cite As Get BibTex

Gaurav Sinha. Efficient Reconstruction of Depth Three Arithmetic Circuits with Top Fan-In Two. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 118:1-118:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.118

Author Details

Gaurav Sinha
  • Adobe Research, Bangalore, India

Acknowledgements

I would like to thank Vineet Nair for helping with organization and presentation of the paper. I would also like to thank Neeraj Kayal and Chandan Saha for helpful comments on an early presentation of this work. Neeraj Kayal shared the simple idea behind proof of Lemma 52 with me. Lastly, I would like to thank Anuja Sharan for proofreading the paper.

References

  1. Eric Allender and Shuichi Hirahara. New insights on the (non-)hardness of circuit minimization and related problems. ACM Trans. Comput. Theory, 11(4), September 2019. URL: https://doi.org/10.1145/3349616.
  2. Dana Angluin. Queries and concept learning. Mach. Learn., 2(4):319-342, April 1988. URL: https://doi.org/10.1023/A:1022821128753.
  3. Boaz Barak, Zeev Dvir, Avi Wigderson, and Amir Yehudayoff. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the Forty-Third Annual ACM Symposium on Theory of Computing, STOC '11, pages 519-528, New York, NY, USA, 2011. Association for Computing Machinery. URL: https://doi.org/10.1145/1993636.1993705.
  4. Vishwas Bhargava, Shubhangi Saraf, and Ilya Volkovich. Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits. CoRR, abs/2105.01751, 2021. URL: http://arxiv.org/abs/2105.01751.
  5. Enrico Carlini. Reducing the number of variables of a polynomial. In Mohamed Elkadi, Bernard Mourrain, and Ragni Piene, editors, Algebraic Geometry and Geometric Modeling, pages 237-247, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg. Google Scholar
  6. Zeev Dvir. Incidence theorems and their applications. Foundations and Trends® in Theoretical Computer Science, 6(4):257-393, 2012. URL: https://doi.org/10.1561/0400000056.
  7. Zeev Dvir and Amir Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. In Proceedings of the Thirty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’05, pages 592-601, New York, NY, USA, 2005. Association for Computing Machinery. URL: https://doi.org/10.1145/1060590.1060678.
  8. Shuhong Gao, Erich Kaltofen, and Alan G.B. Lauder. Deterministic distinct-degree factorization of polynomials over finite fields. Journal of Symbolic Computation, 38(6):1461-1470, 2004. URL: https://doi.org/10.1016/j.jsc.2004.05.004.
  9. Ankit Garg, Neeraj Kayal, and Chandan Saha. Learning sums of powers of low-degree polynomials in the non-degenerate case. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 889-899, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00087.
  10. Shafi Goldwasser and Silvio Micali. Probabilistic encryption. Journal of Computer and System Sciences, 28(2):270-299, 1984. URL: https://doi.org/10.1016/0022-0000(84)90070-9.
  11. Ankit Gupta, Neeraj Kayal, and Satyanarayana V. Lokam. Reconstruction of depth-4 multilinear circuits with top fan-in 2. In Proceedings of the 44th Symposium on Theory of Computing Conference, STOC 2012, New York, NY, USA, May 19 - 22, 2012, pages 625-642, 2012. URL: https://doi.org/10.1145/2213977.2214035.
  12. M. M. Kapranov I. M. Gelfand and A. Zelevinsky. Discriminants, resultants and multidimensional determinants. The Mathematical Gazette, 79(485):439-440, 1995. URL: https://doi.org/10.2307/3618356.
  13. Douglas John Ierardi. The Complexity of Quantifier Elimination in the Theory of an Algebraically Closed Field. PhD thesis, Cornell University, USA, 1989. AAI9001370. Google Scholar
  14. Valentine Kabanets and Jin-Yi Cai. Circuit minimization problem. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, STOC '00, pages 73-79, New York, NY, USA, 2000. Association for Computing Machinery. URL: https://doi.org/10.1145/335305.335314.
  15. Erich Kaltofen. Effective noether irreducibility forms and applications. In Proceedings of the Twenty-third Annual ACM Symposium on Theory of Computing, STOC '91, pages 54-63, New York, NY, USA, 1991. ACM. URL: https://doi.org/10.1145/103418.103431.
  16. Erich Kaltofen and Barry M. Trager. Computing with polynomials given by black boxes for their evaluations: Greatest common divisors, factorization, separation of numerators and denominators. J. Symb. Comput., 9:301-320, 1990. Google Scholar
  17. Zohar S. Karnin and Amir Shpilka. Reconstruction of generalized depth-3 arithmetic circuits with bounded top fan-in. In Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, CCC '09, pages 274-285, Washington, DC, USA, 2009. IEEE Computer Society. URL: https://doi.org/10.1109/CCC.2009.18.
  18. Neeraj Kayal. Derandomizing some algebraic and number-theoretic algorithms. PhD Thesis, IIT Kanpur, January 2006. Google Scholar
  19. Neeraj Kayal. Efficient algorithms for some special cases of the polynomial equivalence problem. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '11, pages 1409-1421, USA, 2011. Society for Industrial and Applied Mathematics. Google Scholar
  20. Neeraj Kayal and Chandan Saha. Reconstruction of non-degenerate homogeneous depth three circuits. In STOC, 2018. Google Scholar
  21. Adam R. Klivans and Amir Shpilka. Learning arithmetic circuits via partial derivatives. In Bernhard Schölkopf and Manfred K. Warmuth, editors, Learning Theory and Kernel Machines, pages 463-476, Berlin, Heidelberg, 2003. Springer Berlin Heidelberg. Google Scholar
  22. Adam R. Klivans and Daniel Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, STOC '01, pages 216-223, New York, NY, USA, 2001. Association for Computing Machinery. URL: https://doi.org/10.1145/380752.380801.
  23. S. Kopparty, S. Saraf, and A. Shpilka. Equivalence of polynomial identity testing and deterministic multivariate polynomial factorization. In Computational Complexity (CCC), 2014 IEEE 29th Conference on, pages 169-180, June 2014. URL: https://doi.org/10.1109/CCC.2014.25.
  24. Daniel Lazard. Solving systems of algebraic equations. SIGSAM Bull., 35(3):11-37, September 2001. URL: https://doi.org/10.1145/569746.569750.
  25. Arjen Lenstra, H. Lenstra, and Lovász László. Factoring polynomials with rational coefficients. Mathematische Annalen, 261, December 1982. URL: https://doi.org/10.1007/BF01457454.
  26. Gary L. Mullen and Daniel Panario. Handbook of Finite Fields. Chapman & Hall/CRC, 1st edition, 2013. Google Scholar
  27. Michael O. Rabin. How to exchange secrets with oblivious transfer, 2005. Harvard University Technical Report 81 talr@watson.ibm.com 12955 received 21 Jun 2005. URL: http://eprint.iacr.org/2005/187.
  28. Nitin Saxena. Progress on polynomial identity testing. Bulletin of the EATCS, 99:49-79, 2009. Google Scholar
  29. Nitin Saxena and C. Seshadhri. Blackbox identity testing for bounded top fanin depth-3 circuits: The field doesn't matter. In Proceedings of the Forty-third Annual ACM Symposium on Theory of Computing, STOC '11, pages 431-440, New York, NY, USA, 2011. ACM. URL: https://doi.org/10.1145/1993636.1993694.
  30. Nitin Saxena and C. Seshadhri. From sylvester-gallai configurations to rank bounds: Improved blackbox identity test for depth-3 circuits. J. ACM, 60(5), October 2013. URL: https://doi.org/10.1145/2528403.
  31. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, October 1980. URL: https://doi.org/10.1145/322217.322225.
  32. Victor Shoup. A fast deterministic algorithm for factoring polynomials over finite fields of small characteristic. In Proceedings of the 1991 International Symposium on Symbolic and Algebraic Computation, ISSAC '91, pages 14-21, New York, NY, USA, 1991. Association for Computing Machinery. URL: https://doi.org/10.1145/120694.120697.
  33. Amir Shpilka. Interpolation of depth-3 arithmetic circuits with two multiplication gates. SIAM J. Comput., 38:2130-2161, 2007. Google Scholar
  34. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends® in Theoretical Computer Science, 5(3–4):207-388, 2010. URL: https://doi.org/10.1561/0400000039.
  35. Gaurav Sinha. Blackbox Reconstruction of Depth Three Circuits with Top Fan-In Two. PhD thesis, California Institute of Technology, Pasadena, CA, USA, 2016. URL: https://doi.org/10.7907/Z92N507D.
  36. Gaurav Sinha. Reconstruction of real depth-3 circuits with top fan-in 2. In Proceedings of the 31st Conference on Computational Complexity, CCC '16, pages 31:1-31:53, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.31.
  37. Ilya Volkovich. A guide to learning arithmetic circuits. In Vitaly Feldman, Alexander Rakhlin, and Ohad Shamir, editors, 29th Annual Conference on Learning Theory, volume 49 of Proceedings of Machine Learning Research, pages 1540-1561, Columbia University, New York, New York, USA, 23-26 June 2016. PMLR. URL: http://proceedings.mlr.press/v49/volkovich16.html.
  38. Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, 3 edition, 2013. URL: https://doi.org/10.1017/CBO9781139856065.
  39. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposiumon on Symbolic and Algebraic Computation, EUROSAM ’79, pages 216-226, Berlin, Heidelberg, 1979. Springer-Verlag. Google Scholar
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