International Symposium on Mathematical Foundations of Computer Science

MFCS 2015: Mathematical Foundations of Computer Science 2015 pp 324-335 | Cite as

The Shifted Partial Derivative Complexity of Elementary Symmetric Polynomials

  • Hervé Fournier
  • Nutan Limaye
  • Meena Mahajan
  • Srikanth Srinivasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9235)

Abstract

We continue the study of the shifted partial derivative measure, introduced by Kayal (ECCC 2012), which has been used to prove many strong depth-4 circuit lower bounds starting from the work of Kayal, and that of Gupta et al. (CCC 2013).

We show a strong lower bound on the dimension of the shifted partial derivative space of the Elementary Symmetric Polynomials of degree d in N variables for \(d < \log N / \log \log N\). This extends the work of Nisan and Wigderson (Computational Complexity 1997), who studied the partial derivative space of these polynomials. Prior to our work, there have been no results on the shifted partial derivative measure of these polynomials. Our result implies a strong lower bound for Elementary Symmetric Polynomials in the homogeneous \({\Sigma \Pi \Sigma \Pi }\) model with bounded bottom fan-in. This strengthens (under our degree assumptions) a lower bound of Nisan and Wigderson who proved the analogous result for homogeneous \({\Sigma \Pi \Sigma }\) model (i.e. \({\Sigma \Pi \Sigma \Pi }\) formulas with bottom fan-in 1).

Our main technical lemma gives a lower bound for the ranks of certain inclusion-like matrices, and may be of independent interest.

References

  1. 1.
    Agrawal, M., Vinay, V.: Arithmetic circuits: a chasm at depth four. In: FOCS, pp. 67–75 (2008)Google Scholar
  2. 2.
    Alon, N.: Perturbed identity matrices have high rank: proof and applications. Comb. Probab. Comput. 18(1–2), 3–15 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Baur, W., Strassen, V.: The complexity of partial derivatives. Theor. Comput. Sci. 22, 317–330 (1983)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Fournier, H., Limaye, N., Malod, G., Srinivasan, S.: Lower bounds for depth 4 formulas computing iterated matrix multiplication. In: Symposium on Theory of Computing, STOC, pp. 128–135 (2014)Google Scholar
  5. 5.
    Gupta, A., Kamath, P., Kayal, N., Saptharishi, R.: Approaching the chasm at depth four. In: Conference on Computational Complexity (CCC) (2013)Google Scholar
  6. 6.
    Hrubes, P., Yehudayoff, A.: Homogeneous formulas and symmetric polynomials. Comput. Complexity 20(3), 559–578 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Kayal, N.: An exponential lower bound for the sum of powers of bounded degree polynomials. Electronic Colloquium on Computational Complexity (ECCC) 19, 81 (2012)Google Scholar
  8. 8.
    Kayal, N., Limaye, N., Saha, C., Srinivasan, S.: An exponential lower bound for homogeneous depth four arithmetic formulas. In: Foundations of Computer Science (FOCS) (2014)Google Scholar
  9. 9.
    Kayal, N., Saha, C., Saptharishi, R.: A super-polynomial lower bound for regular arithmetic formulas. In: STOC, pp. 146–153 (2014)Google Scholar
  10. 10.
    Keevash, P., Sudakov, B.: Set systems with restricted cross-intersections and the minimum rank ofinclusion matrices. SIAM J. Discrete Math. 18(4), 713–727 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Koiran, P.: Arithmetic circuits: the chasm at depth four gets wider. Theor. Comput. Sci. 448, 56–65 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kumar, M., Saraf, S.: The limits of depth reduction for arithmetic formulas: it’s all about the top fan-in. In: STOC, pp. 136–145 (2014)Google Scholar
  13. 13.
    Kumar, M., Saraf, S.: On the power of homogeneous depth 4 arithmetic circuits. In: FOCS, pp. 364–373 (2014)Google Scholar
  14. 14.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, New York (1997)CrossRefMATHGoogle Scholar
  15. 15.
    Nisan, N., Wigderson, A.: Lower bounds on arithmetic circuits via partial derivatives. Comput. Complex. 6(3), 217–234 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Raz, R.: Separation of multilinear circuit and formula size. Theor. Comput. 2(1), 121–135 (2006)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. J. ACM, 56(2) (2009)Google Scholar
  18. 18.
    Razborov, A.: Lower bounds on the size of bounded depth circuits over a complete basis with logical addition. Math. Notes Acad. Sci. USSR 41(4), 333–338 (1987)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Saptharishi, R.: Unified Approaches to Polynomial Identity Testing and Lower Bounds. Ph.D thesis, Chennai Mathematical Institute (2013)Google Scholar
  20. 20.
    Shpilka, A.: Affine projections of symmetric polynomials. J. Comput. Syst. Sci. 65(4), 639–659 (2002)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Shpilka, A., Wigderson, A.: Depth-3 arithmetic circuits over fields of characteristic zero. Comput. Complex. 10(1), 1–27 (2001)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Tavenas, S.: Improved bounds for reduction to depth 4 and 3. In: Mathematical Foundations of Computer Science (MFCS) (2013)Google Scholar
  23. 23.
    Valiant, L.G.: Completeness classes in algebra. In: 11th ACM Symposium on Theory of Computing (STOC), pp. 249–261. New York, NY, USA (1979)Google Scholar
  24. 24.
    Valiant, L.G., Skyum, S., Berkowitz, S., Rackoff, C.: Fast parallel computation of polynomials using few processors. SIAM J. Comput. 12(4), 641–644 (1983)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Wilson, R.M.: A diagonal form for the incidence matrices of \(t\)-subsets vs. \(k\)-subsets. Eur. J. Comb. 11(6), 609–615 (1990)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Hervé Fournier
    • 1
  • Nutan Limaye
    • 2
  • Meena Mahajan
    • 3
  • Srikanth Srinivasan
    • 4
  1. 1.IMJ-PRGUniv Paris DiderotParisFrance
  2. 2.Department of Computer Science and EngineeringIIT BombayMumbaiIndia
  3. 3.The Institute of Mathematical SciencesChennaiIndia
  4. 4.Department of MathematicsIIT BombayMumbaiIndia

Personalised recommendations