Advertisement

Fourier Sparsity of GF(2) Polynomials

  • Hing Yin TsangEmail author
  • Ning Xie
  • Shengyu Zhang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)

Abstract

We study a conjecture called “linear rank conjecture” recently raised in (Tsang et al. [16]), which asserts that if many linear constraints are required to lower the degree of a GF(2) polynomial, then the Fourier sparsity (i.e. number of non-zero Fourier coefficients) of the polynomial must be large. We notice that the conjecture implies a surprising phenomenon that, if the highest degree monomials of a GF(2) polynomial satisfy a certain condition (Specifically, the highest degree monomials do not vanish under a small number of linear restrictions.), then the Fourier sparsity of the polynomial is large regardless of the monomials of lower degrees—whose number is generally much larger than that of the highest degree monomials. We develop a new technique for proving lower bound on the Fourier sparsity of GF(2) polynomials, and apply it to certain special classes of polynomials to showcase the above phenomenon (A full version of this paper is available at http://arxiv.org/abs/1508.02158).

Notes

Acknowledgements

We are indebted to the anonymous reviewers for their detailed helpful comments.

References

  1. 1.
    Ada, A., Fawzi, O., Hatami, H.: Spectral norm of symmetric functions. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds.) APPROX/RANDOM 2012. LNCS, vol. 7408, pp. 338–349. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Bernasconi, A., Codenotti, B.: Spectral analysis of boolean functions as a graph eigenvalue problem. IEEE Trans. Comput. 48(3), 345–351 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cohen, G., Tal, A.: Two structural results for low degree polynomials and applications. ECCC. TR13-145 (2013)Google Scholar
  4. 4.
    Fine, N.: Binomial coefficients modulo a prime. Am. Math. Mon. 54, 589–592 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Göös, M., Pitassi, T., Watson, T.: Deterministic communication vs. partition number. In: Proceedings of the 56th Annual Symposium on Foundations of Computer Science, pp. 1077–1088 (2015)Google Scholar
  6. 6.
    Gopalan, P., O’Donnell, R., Servedio, R., Shpilka, A., Wimme, K.: Testing Fourier dimensionality and sparsity. SIAM J. Comput. 40(4), 1075–1100 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  8. 8.
    Lee, T., Zhang, S.: Composition theorems in communication complexity. In: Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G., Abramsky, S. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 475–489. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Leung, M.L., Li, Y., Zhang, S.: Tight bounds on communication complexity of symmetric XOR functions in one-way and SMP models. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 403–408. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Liu, Y., Zhang, S.: Quantum and randomized communication complexity of XOR functions in the SMP model. ECCC 20, 10 (2013)Google Scholar
  11. 11.
    Lovász, L., Saks, M.E.: Lattices, Möbius functions and communication complexity. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science, pp. 81–90 (1988)Google Scholar
  12. 12.
    Lovett, S.: Communication is bounded by root of rank. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 842–846 (2014)Google Scholar
  13. 13.
    Lovett, S.: Recent advances on the log rank conjecture. Bull. EATCS 112, 18–36 (2014)MathSciNetGoogle Scholar
  14. 14.
    Montanaro, A., Osborne, T.: On the communication complexity of XOR functions (2010). http://arxiv.org/abs/0909.3392v2
  15. 15.
    Shpilka, A., Tal, A., Volk, B.L.: On the structure of boolean functions with small spectral norm. In: Proceedings of the 5th Innovations in Theoretical Computer Science (2014)Google Scholar
  16. 16.
    Tsang, H.Y., Wong, C.H., Xie, N., Zhang, S.: Fourier sparsity, spectral norm, and the log-rank conjecture. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, pp. 658–667 (2013)Google Scholar
  17. 17.
    Yao, A.: Some complexity questions related to distributive computing. In: Proceedings of the 11th Annual ACM Symposium on Theory of Computing, pp. 209–213 (1979)Google Scholar
  18. 18.
    Zhang, Z., Shi, Y.: Communication complexities of symmetric XOR functions. Quant. Inf. Comput. 9(3), 255–263 (2009)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.University of ChicagoChicagoUSA
  2. 2.Florida International UniversityMiamiUSA
  3. 3.The Chinese University of Hong KongHong KongChina

Personalised recommendations