Combinatorica

, Volume 27, Issue 4, pp 439–463

Complexity measures of sign matrices

  • Nati Linial
  • Shahar Mendelson
  • Gideon Schechtman
  • Adi Shraibman
Article

Abstract

In this paper we consider four previously known parameters of sign matrices from a complexity-theoretic perspective. The main technical contributions are tight (or nearly tight) inequalities that we establish among these parameters. Several new open problems are raised as well.

Mathematics Subject Classification (2000)

68Q15 68Q17 46B07 68Q32 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    N. Alon, P. Frankl and V. Rödl: Geometrical realizations of set systems and probabilistic communication complexity, in Proceedings of the 26th Symposium on Foundations of Computer Science, pages 277–280, IEEE Computer Society Press, 1985.Google Scholar
  2. [2]
    N. Alon and J. H. Spencer: The probabilistic method, Wiley, New York, second edition, 2000.MATHGoogle Scholar
  3. [3]
    R. I. Arriaga and S. Vempala: An algorithmic theory of learning: Robust concepts and random projection, in IEEE Symposium on Foundations of Computer Science, pages 616–623, 1999.Google Scholar
  4. [4]
    S. Ben-David, N. Eiron and H. U. Simon: Limitations of learning via embeddings in Euclidean half-spaces, in 14th Annual Conference on Computational Learning Theory, COLT 2001 and 5th European Conference on Computational Learning Theory, EuroCOLT 2001, Amsterdam, The Netherlands, July 2001, Proceedings, volume 2111, pages 385–401, Springer, Berlin, 2001.Google Scholar
  5. [5]
    R. Bhatia: Matrix Analysis, Springer-Verlag, New York, 1997.Google Scholar
  6. [6]
    C. J. C. Burges: A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery 2(2) (1998), 121–167.CrossRefGoogle Scholar
  7. [7]
    J. Forster: A linear lower bound on the unbounded error probabilistic communication complexity, in SCT: Annual Conference on Structure in Complexity Theory, 2001.Google Scholar
  8. [8]
    J. Forster, M. Krause, S. V. Lokam, R. Mubarakzjanov, N. Schmitt and H. U. Simon: Relations between communication complexity, linear arrangements, and computational complexity; in Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science, pages 171–182, 2001.Google Scholar
  9. [9]
    J. Forster, N. Schmitt and H. U. Simon: Estimating the optimal margins of embeddings in Euclidean half spaces, in 14th Annual Conference on Computational Learning Theory, COLT 2001 and 5th European Conference on Computational Learning Theory, EuroCOLT 2001, Amsterdam, The Netherlands, July 2001, Proceedings, volume 2111, pages 402–415, Springer, Berlin, 2001.Google Scholar
  10. [10]
    J. Friedman: A proof of alon’s second eigenvalue conjecture, in Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 720–724, ACM Press, 2003.Google Scholar
  11. [11]
    F. John: Extremum problems with inequalities as subsidiary conditions, Studies and assays presented to R. Courant in his 60th birthday, pages 187–204, 1948.Google Scholar
  12. [12]
    W. B. Johnson and J. Lindenstrauss: Extensions of lipshitz mappings into a Hilbert space, in Conference in modern analysis and probability (New Haven, Conn., 1982), pages 189–206, Amer. Math. Soc., Providence, RI, 1984.Google Scholar
  13. [13]
    J. Kahn, J. Komlós and E. Szemerédi: On the probability that a random ±1-matrix is singular, Journal of the American Mathematical Society 8(1) (1995), 223–240.CrossRefMathSciNetMATHGoogle Scholar
  14. [14]
    B. Kashin and A. Razborov: Improved lower bounds on the rigidity of Hadamard matrices, Mathematical Notes 63(4) (1998), 471–475.CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    E. Kushilevitz and N. Nisan: Communication Complexity, Cambride University Press, 1997.Google Scholar
  16. [16]
    S. V. Lokam: Spectral methods for matrix rigidity with applications to size-depth tradeoffs and communication complexity, in IEEE Symposium on Foundations of Computer Science, pages 6–15, 1995.Google Scholar
  17. [17]
    A. Lobotzky, R. Phillips and P. Sarnak: Ramanujan graphs, Combinatorica 8(3) (1988), 261–277.CrossRefMathSciNetGoogle Scholar
  18. [18]
    G. A. Margulis: Explicit constructions of expanders, Problemy Peredaci Informacii 9(4) (1973), 71–80.MathSciNetMATHGoogle Scholar
  19. [19]
    A. Nilli: On the second eigenvalue of a graph, Discrete Math. 91(2) (1991), 207–210.CrossRefMathSciNetMATHGoogle Scholar
  20. [20]
    N. Nisan and A. Wigderson: On rank vs. communication complexity, in IEEE Symposium on Foundations of Computer Science, pages 831–836, 1994.Google Scholar
  21. [21]
    R. Paturi and J. Simon: Probabilistic communication complexity, Journal of Computer and System Sciences 33 (1986), 106–123.CrossRefMathSciNetMATHGoogle Scholar
  22. [22]
    G. Pisier: Factorization of linear operators and geometry of Banach spaces, volume 60 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986.Google Scholar
  23. [23]
    P. Pudlák and V. Rödl: Some combinatorial-algebraic problems from complexity theory, Discrete Mathematics 136 (1994), 253–279.CrossRefMathSciNetMATHGoogle Scholar
  24. [24]
    M. A. Shokrollahi, D. A. Spielman and V. Stemann: A remark on matrix rigidity, Information Processing Letters 64(6) (1997), 283–285.CrossRefMathSciNetGoogle Scholar
  25. [25]
    M. Talagrand: Concentration of measures and isoperimetric inequalities in product spaces, Publications Mathematiques de l’I.H.E.S. 81 (1996), 73–205.CrossRefGoogle Scholar
  26. [26]
    T. Tao and V. Vu: On the singularity probability of random Bernoulli matrices, Journal of the American Mathematical Society 20(3) (2007), 603–628.CrossRefMathSciNetMATHGoogle Scholar
  27. [27]
    N. Tomczak-Jaegermann: Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, 1989.MATHGoogle Scholar
  28. [28]
    L. G. Valiant: Graph-theoretic arguments in low level complexity, in Proc. 6th MFCS, volume 53, pages 162–176. Springer-Verlag LNCS, 1977.MathSciNetGoogle Scholar
  29. [29]
    V. N. Vanik: The Nature of Statistical Learning Theory, Springer-Verlag, New York, 1999.Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Nati Linial
    • 1
  • Shahar Mendelson
    • 2
    • 3
  • Gideon Schechtman
    • 4
  • Adi Shraibman
    • 4
  1. 1.School of Computer Science and EngineeringHebrew UniversityJerusalemIsrael
  2. 2.Centre for Mathematics and its ApplicationsThe Australian National UniversityCanberraAustralia
  3. 3.Department of MathematicsTechnion I.I.THaifaIsrael
  4. 4.Department of MathematicsWeizmann InstituteRehovotIsrael

Personalised recommendations