International Conference on Algorithmic Learning Theory

Algorithmic Learning Theory pp 224-238 | Cite as

Subsampling in Smoothed Range Spaces

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9355)


We consider smoothed versions of geometric range spaces, so an element of the ground set (e.g. a point) can be contained in a range with a non-binary value in [0, 1]. Similar notions have been considered for kernels; we extend them to more general types of ranges. We then consider approximations of these range spaces through \(\varepsilon \)-nets and \(\varepsilon \)-samples (aka \(\varepsilon \)-approximations). We characterize when size bounds for \(\varepsilon \)-samples on kernels can be extended to these more general smoothed range spaces. We also describe new generalizations for \(\varepsilon \)-nets to these range spaces and show when results from binary range spaces can carry over to these smoothed ones.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fitzgibbon, A., Bergamo, A., Torresani, L.: Picodes: learning a compact code for novel-category recognition. In: NIPS (2011)Google Scholar
  2. 2.
    Alon, N., Ben-David, S., Cesa-Bianchi, N., Haussler, D.: Scale-sensitive dimensions, uniform convergence, and learnability. Journal of ACM 44, 615–631 (1997)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Aronov, B., Ezra, E., Sharir, M.: Small size \(\varepsilon \)-nets for axis-parallel rectangles and boxes. Siam Journal of Computing 39, 3248–3282 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Beck, J.: Irregularities of distribution I. Acta Mathematica 159, 1–49 (1987)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bern, M., Eppstein, D.: Worst-case bounds for subadditive geometric graphs. In: SOCG (1993)Google Scholar
  6. 6.
    Chazelle, B.: The Discrepancy Method. Cambridge (2000)Google Scholar
  7. 7.
    Chazelle, B., Matousek, J.: On linear-time deterministic algorithms for optimization problems in fixed dimensions. J. Algorithms 21, 579–597 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge (2009)Google Scholar
  9. 9.
    Edmonds, J.: Paths, trees, and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Har-Peled, S., Kaplan, H., Sharir, M., Smorodinksy, S.: \(\varepsilon \)-nets for halfspaces revisited. Technical report (2014). arXiv:1410.3154
  11. 11.
    Haussler, D., Welzl, E.: epsilon-nets and simplex range queries. Disc. & Comp. Geom. 2, 127–151 (1987)CrossRefMATHGoogle Scholar
  12. 12.
    Joshi, S., Kommaraju, R.V., Phillips, J.M., Venkatasubramanian, S.: Comparing distributions and shapes using the kernel distance. In: SOCG (2011)Google Scholar
  13. 13.
    Larsen, K.G.: On range searching in the group model and combinatorial discrepancy. In: FOCS (2011)Google Scholar
  14. 14.
    Li, Y., Long, P.M., Srinivasan, A.: Improved bounds on the samples complexity of learning. J. Comp. and Sys. Sci. 62, 516–527 (2001)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Matoušek, J.: Tight upper bounds for the discrepancy of halfspaces. Discrete & Computational Geometry 13, 593–601 (1995)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Matoušek, J.: Geometric Discrepancy. Springer (1999)Google Scholar
  17. 17.
    Matoušek, J., Seidel, R., Welzl, E.: How to net a lot with little: small \(\varepsilon \)-nets for disks and halfspaces. In: SOCG (1990)Google Scholar
  18. 18.
    Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley, Wiley-Interscience series in discrete mathematics and optimization (1995)Google Scholar
  19. 19.
    Pach, J., Tardos, G.: Tight lower bounds for the size of epsilon-nets. Journal of American Mathematical Society 26, 645–658 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Phillips, J.M.: Algorithms for \(\varepsilon \)-approximations of terrains. In: Automata, Languages and Programming, pp. 447–458. Springer (2008)Google Scholar
  21. 21.
    Phillips, J.M.: Eps-samples for kernels. In: SODA (2013)Google Scholar
  22. 22.
    Pollard, D.: Emperical processes: theory and applications. In: NSF-CBMS REgional Confernece Series in Probability and Statistics (1990)Google Scholar
  23. 23.
    Pyrga, E., Ray, S.: New existence proofs \(\varepsilon \)-nets. In: SOCG (2008)Google Scholar
  24. 24.
    Vapnik, V.: Inductive principles of the search for empirical dependencies. In: COLT (1989)Google Scholar
  25. 25.
    Vapnik, V., Chervonenkis, A.: On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications 16, 264–280 (1971)CrossRefMATHGoogle Scholar
  26. 26.
    Varadarajan, K.R.: A divide-and-conquer algorithm for min-cost perfect matching in the plane. In: FOCS (1998)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of UtahSalt Lake CityUSA

Personalised recommendations