On the Vapnik-Chervonenkis dimension of computer programs which use transcendental elementary operations



We exhibit upper bounds for the Vapnik-Chervonenkis (VC) dimension of a wide family of concept classes that are defined by algorithms using analytic Pfaffian functions. We give upper bounds on the VC dimension of concept classes in which the membership test for whether an input belongs to a concept in the class can be performed either by a computation tree or by a circuit with sign gates containing Pfaffian functions as operators. These new bounds are polynomial both in the height of the tree and in the depth of the circuit. As consequence we obtain polynomial VC dimension not also for classes of concepts whose membership test can be defined by polynomial time algorithms but also for those defined by well-parallelizable sequential exponential time algorithms.


Concept learning Vapnik–Chervonenkis dimension Khovanskii bounds Pfaffian functions Parallel computation Formula size 

Mathematics Subject Classifications (2000)

68T05 58A17 


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  1. 1.
    Aldaz, M., Heintz, J., Matera, G., Montaña, J.L., Pardo, L.M.: Combinatorial hardness proofs for polynomial evaluation (extended abstract). In: Mathematical Foundations of Computer science, 1998 (Brno). Lecture Notes in Computer Science, vol. 1450, pp. 167–175. Springer, Berlin (1998)CrossRefGoogle Scholar
  2. 2.
    Ben-Or, M.: Lower bounds for algebraic computation trees. In: Proc. of the 15th Annual ACM Symp. on Theor. Comput., pp. 80–86. ACM, New York (1983)Google Scholar
  3. 3.
    Blumer, A., Ehrenfeucht, A., Haussler, D., Warmuth, M.K.: Learnability and the Vapnik-Chervonenkis dimension. J. ACM 36, 929–965 (1989)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Gabrielov, A.N., Vorobjov, N.: Complexity of computations with Pfaffian and Noetherian functions. In: Normal Forms, Bifurcations and Finiteness Problems in Differential Equations. Kluwer, Dordrecht (2004)Google Scholar
  5. 5.
    Gabrielov, A., Vorobjov, N., Zell, T.: Betti numbers of semialgebraic and sub-Pfaffian sets. J. Lond. Math. Soc. 69, 27–43 (2003)CrossRefMathSciNetGoogle Scholar
  6. 6.
    von zur Gathen, J.: Parallel arithmetic computations, a survey. In: Proc. Math. Found. Comput. Sci. 13th. Lecture Notes in Computer Science, vol. 233, pp. 93–112. Springer, Berlin (1986)Google Scholar
  7. 7.
    Goldberg, P., Jerrum, M.: Bounding the Vapnik-Chervonenkis dimension of concept classes parametrizes by real numbers. Mach. Learn. 18, 131–148 (1995)MATHGoogle Scholar
  8. 8.
    Grigor’ev, D., Vorobjov, N.: Complexity lower bounds for computation trees with elementary transcendental function gates. Theor. Comput. Sci. 157, 185–214 (1996)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Karpinski, M., Macintyre, A.: Polynomial bounds for VC dimension of sigmoidal and general Pffafian neural networks. J. Comput. Syst. Sci. 54, 169–176 (1997)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Khovanskii, A.: On a class of systems of transcendental equations. Sov. Math. Dokl. 22, 762–765 (1980)Google Scholar
  11. 11.
    Khovanskii, A.: Fewnomials. In: AMS Transl. Math. Monographs, vol. 88. AMS, Providence (1991)Google Scholar
  12. 12.
    Milnor, J.: On the Betti numbers of real varieties. Proc. Am. Math. Soc. 15, 275–280 (1964)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Montaña, J.L., Pardo, L.M.: Lower bounds for arithmetic networks. Appl. Algebra Eng. Commun. Comput. 4, 1–24 (1993)MATHCrossRefGoogle Scholar
  14. 14.
    Montaña, J.L., Pardo, L.M., Callau, M.: VC dimension bounds for analytic algebraic computations. In: Proc. Computing and Combinatorics, 14 Annual Intenational Conference. Lecture Notes in Computer Science, vol. 5092, pp. 62–71. Springer, Berlin (2008)Google Scholar
  15. 15.
    Oleinik, O.A.: Estimates of the Betti numbers of real algebraic hypersurfaces. Mat. Sb. N.S. 28, 635–640 (1951) (in Russian)MathSciNetGoogle Scholar
  16. 16.
    Oleinik, O.A., Petrovsky, I.B.: On the topology of real algebraic surfaces. Izv. Akad. Nauk SSSR (in Trans. Am. Math. Soc.) 1, 399–417 (1962)Google Scholar
  17. 17.
    Thom, R.: Sur l’Homologie des Variétés Algébriques Réelles. In: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), pp. 255–265. Princeton University Press, Princeton (1965)Google Scholar
  18. 18.
    Valiant, L.G.: A theory of the learneable. Commun. ACM 27, 1134–1142 (1984)MATHCrossRefGoogle Scholar
  19. 19.
    Vapnik, V., Chervonenkis, A.: On the uniform convergence of relative frequencies of events to their probabilities. Theory Probab. Appl. 16, 264–280 (1971)MATHCrossRefGoogle Scholar
  20. 20.
    Warren, H.E.: Lower bounds for approximation by non linear manifolds. Trans. Am. Math. Soc. 133, 167–178 (1968)MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Depto. de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain

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