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Approximating the volume of general Pfaffian bodies

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Structures in Logic and Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1261))

Abstract

We introduce a new method of approximating volume (and integrals) for a vast number of geometric bodies defined by boolean combinations of Pfaffian conditions. The method depends on the VC Dimension of the underlying classes of bodies. The resulting approximation algorithms are quite different in spirit from previously known methods, and give randomized solutions even for such seemingly intractable problems of statistical physics as computing the volume of sets defined by the systems of exponential (or more generally Pfaffian) inequalities.

Research partially supported by the DFG Grant KA 673/4-1, and by ESPRIT BR Grants 7097 and EC-US 030.

Research supported in part by a Senior Research Fellowship of the SERC.

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References

  1. M. Anthony and N. Biggs, Computational Learning Theory: An Introduction, Cambridge University Press, 1992.

    Google Scholar 

  2. M. Anthony and J. Shawe-Taylor, A Result of Vapnik with Applications, Discrete Applied Math. 47 (1993), pp. 207–217.

    Google Scholar 

  3. A. Blumer, A. Ehrenfeucht, D. Haussler and M. Warmuth, Learnability and the Vapnik-Chervonenkis Dimension, J. ACM 36 (1990), pp. 929–965.

    Google Scholar 

  4. A. Borodin, P. Tiwari, On the Decidability of Sparse Univariate Polynomial Interpolation, Proc. 22nd ACM STOC (1990), pp. 535–545.

    Google Scholar 

  5. F. Delon, Définissabilité avec Paramètres Extérieurs dans Qinp et R, Proc. AMS 106 (1989), pp. 193–198.

    Google Scholar 

  6. J. Denef, The Rationality of the Poincare Series associated to the p-adic Points on a Variety, Invent. Math. 77 (1984), pp. 1–23.

    Google Scholar 

  7. J. Denef and L.P.D. van den Dries, p-adic and real subanalytic Sets, Annals of Mathematics 128 (1988), 79–138.

    Google Scholar 

  8. L. van den Dries, Tame Topology and o-Minimal Structures, preprint, University of Illinois, Urbana, 1992; to appear as a book.

    Google Scholar 

  9. L. van den Dries, D. Haskell, H.D. Macpherson, On Dimensional p-adic Subanalytic Sets, to appear.

    Google Scholar 

  10. L. van den Dries, A. Macintyre and D. Marker, The Elementary Theory of Restricted Analytic Fields with Exponentation, Annals of Mathematics 140 (1994), pp 183–205.

    Google Scholar 

  11. L. van den Dries, C. Miller, Geometric Categories and o-Minimal Theories, Duke Journal 84 (1996), 497–540.

    Google Scholar 

  12. M. E. Dyer and M. Frieze, On the Complexity of Computing the Volume of a Polyhedron, SIAM J. Comput. 17 (1988), pp. 967–974.

    Google Scholar 

  13. M. Dyer, A. Frieze and R. Kannan, A Random Polynomial-Time Algorithm for Approximating the Volume of Convex Bodies, J. ACM 83 (1991), pp. 1–17.

    Google Scholar 

  14. P. Goldberg and M. Jerrum, Bounding the Vapnik Chervonenkis Dimension of Concept Classes Parametrized by Real Numbers. Machine Learning, 1995. A preliminary version appeared in Proc. 6th ACM Workshop on Computational Learning Theory, pp. 361–369, 1993.

    Google Scholar 

  15. P. Halmos, Measure Theory, Chelsey-New York, 1950.

    Google Scholar 

  16. G. H. Hardy, Properties of Logarithmic-Exponential Functions, Proc. London Math. Soc. 10 (1912), pp. 54–90.

    Google Scholar 

  17. D. Haussler, Decision Theoretic Generalizations of the PAC Model for Neural Net and other Learning Applications, Information and Computation 100, (1992), pp. 78–150.

    Google Scholar 

  18. J. Hertz, A. Krogh and R. G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley, 1991.

    Google Scholar 

  19. M. W. Hirsch, Differential Topology, Springer-Verlag, 1976.

    Google Scholar 

  20. M. Karpinski and A. Macintyre, Bounding VC Dimension for Neural Networks: Progress and Prospects (Invited Lecture), Proc. EuroCOLT'95, Lecture Notes in Artificial Intelligence Vol. 904, Springer-Verlag, 1995, pp. 337–341.

    Google Scholar 

  21. M. Karpinski and A. Macintyre, Polynomial Bounds for VC Dimension of Sigmoidal Neural Networks, Proc. 27th ACM STOC (1995), pp. 200–208.

    Google Scholar 

  22. M. Karpinski and A. Macintyre, Polynomial Bounds for VC Dimension of Sigmoidal and General Pfaffian Neural Networks, J. Comput. Syst. Sci. 54 (1997), to appear.

    Google Scholar 

  23. M. Karpinski, A. Macintyre, Approximating Volumes and Integrals in o-Minimal and p-Minimal Theories, manuscript in preparation.

    Google Scholar 

  24. M. Karpinski and T. Werther, VC Dimension and Uniform Learnability of Sparse Polynomials and Rational Functions, SIAM J. Computing 22 (1993), pp 1276–1285.

    Google Scholar 

  25. A. G. Khovanski, Fewnomials, American Mathematical Society, Providence, R.I., 1991.

    Google Scholar 

  26. J. Knight, A. Pillay and C. Steinhorn, Definable Sets and Ordered Structures II, Trans. American Mathematical Society 295 (1986), pp. 593–605.

    Google Scholar 

  27. P. Koiran, Approximating the Volume of Definable Sets, Proc. 36th IEEE FOCS (1995), pp. 258–265.

    Google Scholar 

  28. P. Koiran and E. D. Sontag, Neural Networks with Quadratic VC Dimension to appear in Advances in Neural Information Processing Systems (NIPS '95), 1995.

    Google Scholar 

  29. M. C. Laskowski, Vapnik-Chervonenkis Classes of Definable Sets, J. London Math. Society 45 (1992), pp 377–384.

    Google Scholar 

  30. L. Lovasz and M. Simonovits, The Mixing Rate of Markov Chains, an Isoperimetric Inequality, and Computing the Volume, Proc. 31 IEEE FOCS (1990), pp. 346–355.

    Google Scholar 

  31. W. Maass, Perspectives of Current Research about the Complexity of Learning on Neural Nets, in: Theoretical Advances in Neural Computation and Learning, V. P. Roychowdhury, K. Y. Siu, A. Orlitsky (Editors), Kluwer Academic Publishers, 1994, pp. 295–336.

    Google Scholar 

  32. W. Maass, Bounds for the Computational Power and Learning Complexity of Analog Neural Nets, Proc. 25th ACM STOC (1993), pp. 335–344.

    Google Scholar 

  33. W. Maass, G. Schnitger and E. D. Sontag, On the Computational Power of Sigmoidal versus Boolean Threshold Circuits, Proc. 32nd IEEE FOCS (1991), pp. 767–776.

    Google Scholar 

  34. A. J. Macintyre and E. D. Sontag, Finiteness results for Sigmoidal Neural Networks, Proc. 25th ACM STOC (1993), pp. 325–334.

    Google Scholar 

  35. J. Milnor, On the Betti Numbers of Real Varieties, Proc. of the American Mathematical Society 15 (1964), pp 275–280.

    Google Scholar 

  36. J. Milnor, Topology from the Differentiable Viewpoint, Univ.Press, Virginia, 1965.

    Google Scholar 

  37. B. Pouzat, Groupes stables, avec types géneriques réguliers, J. S. L. 48 (1983), pp. 339–355.

    Google Scholar 

  38. J. Renegar, On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Parts I, II, III, J. of Symb. Comput. 3 (1992), pp. 255–352.

    Google Scholar 

  39. A. Sard, The Measure of the Critical Points of Differentiable Maps, Bull. Amer. Math. Soc. 48 (1942), pp. 883–890.

    Google Scholar 

  40. J. Shawe-Taylor, Sample Sizes for Sigmoidal Neural Networks, Preprint, University of London, 1994, to appear in Proc. ACM COLT, 1995.

    Google Scholar 

  41. E. D. Sontag, Feedforward Nets for Interpolation and Classification, J. Comp. Syst. Sci. 45 (1992), pp. 20–48.

    Google Scholar 

  42. V. Vapnik, Estimation of Dependencies Based on Empirical Data, Springer Series in Statistics, 1982.

    Google Scholar 

  43. H. E. Warren, Lower Bounds for Approximation by Non-linear Manifolds, Trans. of the AMS 133 (1968), pp. 167–178.

    Google Scholar 

  44. A. J. Wilkie, Model Completeness Results of Restricted Pfaffian Functions and the Exponential Function; to appear in Journal of the AMS, 1994.

    Google Scholar 

  45. A.J. Wilkie, Handwritten manuscript, Oxford 1996.

    Google Scholar 

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Authors and Affiliations

  1. Dept. of Computer Science, University of Bonn, 53117, Bonn

    Marek Karpinski

  2. International Computer Science Institute, Berkeley

    Marek Karpinski

  3. Mathematical Institute, University of Oxford, 0X1 3LB, Oxford

    Angus Macintyre

Authors
  1. Marek Karpinski
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  2. Angus Macintyre
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Jan Mycielski Grzegorz Rozenberg Arto Salomaa

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Karpinski, M., Macintyre, A. (1997). Approximating the volume of general Pfaffian bodies. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds) Structures in Logic and Computer Science. Lecture Notes in Computer Science, vol 1261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63246-8_10

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  • DOI: https://doi.org/10.1007/3-540-63246-8_10

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  • Print ISBN: 978-3-540-63246-7

  • Online ISBN: 978-3-540-69242-3

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