Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

Authors Rafael Oliveira, Amir Shpilka, Ben Lee Volk



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Rafael Oliveira
Amir Shpilka
Ben Lee Volk

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Rafael Oliveira, Amir Shpilka, and Ben Lee Volk. Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 304-322, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.CCC.2015.304

Abstract

In this paper we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds we obtain lower bounds for these models. For depth-3 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 2*delta/3))). This implies a lower bound of exp(~Omega(n^(1/2))) for depth-3 multilinear formulas, for some explicit polynomial. For depth-4 multilinear formulas, of size exp(n^delta), we give a hitting set of size exp(~O(n^(2/3 + 4*delta/3)). This implies a lower bound of exp(~Omega(n^(1/4))) for depth-4 multilinear formulas, for some explicit polynomial. A regular formula consists of alternating layers of +,* gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp(n^(1-delta)), for regular depth-d multilinear formulas of size exp(n^delta), where delta = O(1/sqrt(5)^d)). This result implies a lower bound of roughly exp(~Omega(n^(1/sqrt(5)^d))) for such formulas. We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known. Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas we go straight to read-once algebraic branching programs).
Keywords
  • Arithmetic Circuits
  • Derandomization
  • Polynomial Identity Testing

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References

  1. M. Agrawal. Proving lower bounds via pseudo-random generators. In Proceedings of the 25th FSTTCS, volume 3821 of LNCS, pages 92-105, 2005. Google Scholar
  2. M. Agrawal, R. Gurjar, A. Korwar, and N. Saxena. Hitting-sets for ROABP and sum of set-multilinear circuits. Electronic Colloquium on Computational Complexity (ECCC), 21:85, 2014. Google Scholar
  3. M. Agrawal, N. Kayal, and N. Saxena. Primes is in P. Annals of Mathematics, 160(2):781-793, 2004. Google Scholar
  4. M. Agrawal, C. Saha, R. Saptharishi, and N. Saxena. Jacobian hits circuits: hitting-sets, lower bounds for depth-d occur-k formulas & depth-3 transcendence degree-k circuits. In STOC, pages 599-614, 2012. Google Scholar
  5. M. Agrawal, C. Saha, and N. Saxena. Quasi-polynomial hitting-set for set-depth-Δ formulas. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 321-330, 2013. Google Scholar
  6. M. Agrawal and V. Vinay. Arithmetic circuits: A chasm at depth four. In Proceedings of the 49th Annual FOCS, pages 67-75, 2008. Google Scholar
  7. N. Alon and J. H. Spencer. The probabilistic method. J. Wiley, 3 edition, 2008. Google Scholar
  8. M. Anderson, D. van Melkebeek, and I. Volkovich. Derandomizing polynomial identity testing for multilinear constant-read formulae. In Proceedings of the 26th Annual CCC, pages 273-282, 2011. Google Scholar
  9. R. A. DeMillo and R. J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. Google Scholar
  10. Z. Dvir and A. Shpilka. Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits. SIAM J. on Computing, 36(5):1404-1434, 2006. Google Scholar
  11. Z. Dvir, A. Shpilka, and A. Yehudayoff. Hardness-randomness tradeoffs for bounded depth arithmetic circuits. SIAM J. on Computing, 39(4):1279-1293, 2009. Google Scholar
  12. M. A. Forbes, R. Saptharishi, and A. Shpilka. Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 867-875, 2014. Google Scholar
  13. M. A. Forbes and A. Shpilka. On identity testing of tensors, low-rank recovery and compressed sensing. In Proceedings of the 44th annual STOC, pages 163-172, 2012. Google Scholar
  14. M. A. Forbes and A. Shpilka. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 243-252, 2013. Google Scholar
  15. A. Gupta, P. Kamath, N. Kayal, and R. Saptharishi. Arithmetic circuits: A chasm at depth three. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, pages 578-587, 2013. Google Scholar
  16. J. Heintz and C. P. Schnorr. Testing polynomials which are easy to compute (extended abstract). In Proceedings of the 12th annual STOC, pages 262-272, 1980. Google Scholar
  17. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. In Proceedings of the 35th Annual STOC, pages 355-364, 2003. Google Scholar
  18. V. Kabanets and R. Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity, 13(1-2):1-46, 2004. Google Scholar
  19. Z. S. Karnin, P. Mukhopadhyay, A. Shpilka, and I. Volkovich. Deterministic identity testing of depth-4 multilinear circuits with bounded top fan-in. SIAM J. Comput., 42(6):2114-2131, 2013. Google Scholar
  20. Z. S. Karnin and A. Shpilka. Black box polynomial identity testing of generalized depth-3 arithmetic circuits with bounded top fan-in. Combinatorica, 31(3):333-364, 2011. Google Scholar
  21. N. Kayal, C. Saha, and R. Saptharishi. A super-polynomial lower bound for regular arithmetic formulas. In Symposium on Theory of Computing, STOC 2014, pages 146-153, 2014. Google Scholar
  22. N. Kayal and S. Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Proceedings of the 50th Annual FOCS, pages 198-207, 2009. Google Scholar
  23. N. Kayal and N. Saxena. Polynomial identity testing for depth 3 circuits. Computational Complexity, 16(2):115-138, 2007. Google Scholar
  24. P. Koiran. Arithmetic circuits: the chasm at depth four gets wider. CoRR, abs/1006.4700, 2010. Google Scholar
  25. N. Nisan. Lower bounds for non-commutative computation. In Proceedings of the 23rd Annual STOC, pages 410-418, 1991. Google Scholar
  26. N. Nisan and A. Wigderson. Lower bound on arithmetic circuits via partial derivatives. Computational Complexity, 6:217-234, 1996. Google Scholar
  27. R. Oliveira, A. Shpilka, and B. L. Volk. Subexponential size hitting sets for bounded depth multilinear formulas. Electronic Colloquium on Computational Complexity (ECCC), 21:157, 2014. Google Scholar
  28. R. Raz and A. Shpilka. Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1-19, 2005. Google Scholar
  29. R. Raz and A. Yehudayoff. Lower bounds and separations for constant depth multilinear circuits. Computational Complexity, 18(2):171-207, 2009. Google Scholar
  30. S. Saraf and I. Volkovich. Black-box identity testing of depth-4 multilinear circuits. In Proceedings of the 43rd annual STOC, pages 421-430, 2011. Google Scholar
  31. N. Saxena and C. Seshadhri. Blackbox identity testing for bounded top fanin depth-3 circuits: the field doesn't matter. In Proceedings of the 43rd Annual STOC, pages 431-440, 2011. Google Scholar
  32. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. Google Scholar
  33. A. Shpilka and I. Volkovich. Read-once polynomial identity testing. In Proceedings of the 40th Annual STOC, pages 507-516, 2008. Google Scholar
  34. A. Shpilka and I. Volkovich. Improved polynomial identity testing for read-once formulas. In APPROX-RANDOM, pages 700-713, 2009. Google Scholar
  35. A. Shpilka and A. Yehudayoff. Arithmetic circuits: A survey of recent results and open questions. Foundations and Trends in Theoretical Computer Science, 5(3-4):207-388, 2010. Google Scholar
  36. S. Tavenas. Improved bounds for reduction to depth 4 and depth 3. In Mathematical Foundations of Computer Science 2013 - 38th International Symposium, MFCS 2013, pages 813-824, 2013. Google Scholar
  37. L. G. Valiant, S. Skyum, S. Berkowitz, and C. Rackoff. Fast parallel computation of polynomials using few processors. SIAM J. on Computing, 12(4):641-644, November 1983. Google Scholar
  38. R. Zippel. Probabilistic algorithms for sparse polynomials. In EUROSAM, pages 216-226, 1979. Google Scholar
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