Spectral Norm in Learning Theory: Some Selected Topics

  • Hans Ulrich Simon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4264)


In this paper, we review some known results that relate the statistical query complexity of a concept class to the spectral norm of its correlation matrix. Since spectral norms are widely used in various other areas, we are then able to put statistical query complexity in a broader context. We briefly describe some non-trivial connections to (seemingly) different topics in learning theory, complexity theory, and cryptography. A connection to the so-called Hidden Number Problem, which plays an important role for proving bit-security of cryptographic functions, will be discussed in somewhat more detail.


Learn Theory Incidence Matrix Concept Class Spectral Norm Target Concept 
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  1. 1.
    Ben-David, S., Itai, A., Kushilevitz, E.: Learning by distances. Information and Computation 117(2), 240–250 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Blum, A., Furst, M., Jackson, J., Kearns, M., Mansour, Y., Rudich, S.: Weakly learning DNF and characterizing statistical query learning using Fourier analysis. In: Proceedings of the 26th Annual Symposium on Theory of Computing, pp. 253–263 (1994)Google Scholar
  3. 3.
    Blum, A., Kalai, A., Wasserman, H.: Noise-tolerant learning, the parity problem, and the statistical query model. Journal of the Association on Computing Machinery 50(4), 506–519 (2003)MathSciNetGoogle Scholar
  4. 4.
    Boneh, D., Venkatesan, R.: Hardness of Computing the Most Significant Bits of Secret Keys in Diffie-Hellman and Related Schemes. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 129–142. Springer, Heidelberg (1996)Google Scholar
  5. 5.
    Forster, J.: A linear lower bound on the unbounded error communication complexity. Journal of Computer and System Sciences 65(4), 612–625 (2002)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Forster, J., Krause, M., Lokam, S.V., Mubarakzjanov, R., Schmitt, N., Simon, H.U.: Relations between communication complexity, linear arrangements, and computational complexity. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, p. 171. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  7. 7.
    Kearns, M.: Efficient noise-tolerant learning from statistical queries. Journal of the Association on Computing Machinery 45(6), 983–1006 (1998)MATHMathSciNetGoogle Scholar
  8. 8.
    Kiltz, E.: A useful primitive to prove security of every bit and about hard core predicates and universal hash functions. In: Proceedings of the 14th International Symposium on Fundamentals of Computation Theory, pp. 388–392 (2001)Google Scholar
  9. 9.
    Kiltz, E., Simon, H.U.: Unpublished Manuscript about the Hidden Number ProblemGoogle Scholar
  10. 10.
    Kiltz, E., Simon, H.U.: Threshold circuit lower bounds on cryptographic functions. Journal of Computer and System Sciences 71(2), 185–212 (2005)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Krause, M., Waack, S.: Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in. Mathematical System Theory 28(6), 553–564 (1995)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Nguyên, P.Q., Stern, J.: The two faces of lattices in cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, p. 146. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  13. 13.
    Paturi, R., Simon, J.: Probabilistic communication complexity. Journal of Computer and System Sciences 33(1), 106–123 (1986)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Valiant, L.G.: A theory of the learnable. Communications of the ACM 27(11), 1134–1142 (1984)MATHCrossRefGoogle Scholar
  15. 15.
    Vasco, M.I.G., Shparlinski, I.E.: On the security of Diffie–Hellman bits. In: Proceedings of the Workshop on Cryptography and Computational Number Theory, pp. 331–342 (2000)Google Scholar
  16. 16.
    Yang, K.: On learning correlated boolean functions using statistical queries (Extended abstract). In: Abe, N., Khardon, R., Zeugmann, T. (eds.) ALT 2001. LNCS, vol. 2225, p. 59. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Yang, K.: New lower bounds for statistical query learning. In: Kivinen, J., Sloan, R.H. (eds.) COLT 2002. LNCS, vol. 2375, p. 229. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  18. 18.
    Yang, K., Blum, A.: On statistical query sampling and nmr quantum computing. In: Proceedings of the 18th Annual Conference on Computational Complexity, pp. 194–208 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Hans Ulrich Simon
    • 1
  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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