Universal ε-approximators for integrals

Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Volumn , Issue , 2010, Pages 598-607

  Langberg, Michael ^a   Schulman, Leonard J ^b  

^a OPEN UNIVERSITY OF ISRAEL   (Israel)

^b CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHMS;

APPROXIMATORS; CARDINALITIES; COMPUTATIONAL LEARNING THEORY; DATA CLASSIFICATION; NON NEGATIVES; PROBABILITY MEASURES; REAL FUNCTIONS; VC DIMENSION;

COMPUTATION THEORY;

EID: 77951686690     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1137/1.9781611973075.50     Document Type: Conference Paper

References (38)

1. Pankaj Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Geometric approximation via coresets. In Current Trends in Combinatorial and Computational Geometry. Cambridge University Press, 2005.
2. N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler. Scale-sensitive dimensions, uniform convergence, and learnability. J. ACM, 44(4):615-631, July 1997.
3. N. Alon and B. Sudakov. On two segmentation problems. J. of Algorithms, 33:173-184, 1999.
4. P. L. Bartlett and P. M. Long. Prediction, learning, uniform convergence, and scale-sensitive dimensions. J. Comput. System Sci., 56:174-190, 1998.
5. P. L. Bartlett, P. M. Long, and R. C. Williamson. Fat-shattering and the learnability of real-valued functions. J. Comput. System Sci., 52:434-452, 1996.
6. S. Ben-David, N. Cesa-Bianchi, D. Haussler, and P. M. Long. Characterizations of learnability for classes of {0, . . . ,n}-valued functions. J. of Comput. Syst. Sci., 50(1):74-86, 1995.
7. S. Ben-David, N. Cesa-Bianchi, and P. M. Long. Characterizations of learnability for classes of {0, . . . , n}-valued functions. In Proc. 5'th Annual ACM COLT, pages 333-340, 1992.
8. A, Blumer, A. Ehrenfeucht, D. Haussler, and M. K. Warmuth. Learnability and the Vapnik-Chervonenkis dimension. J. ACM, 36(4):929-965, October 1989.
9. B. Chazelle. The Discrepancy Method. Cambridge U. Press, 2000.
10. K. Chen. On coresets for k-median and k-means clustering in metric and Euclidean spaces and their applications. In Proc. 17th SODA, pages 1177-1185, 2006.
11. W. Fernandez de la Vega and C. Kenyon. A randomized approximation scheme for metric max-cut. In 39th Ann. Symp. on Foundations of Computer Science, pages 468-471. IEEE, 1998.
12. R. M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. J. Approximation Theory, 10(3):227-236, 1974.
13. R. M. Dudley, E. Giné, and J. Zinn. Uniform and universal Glivenko-Cantelli classes. J. Theoretical Probability, 4(3):485-510, 1991.
14. M. Effros and L. J. Schulman. Deterministic clustering with data nets. http://eccc.hpi-web.de/eccc-reports/2004/TR04-050/index.html, 2004. Technical Report TR04-085, Elec. Colloq. Comp. Complexity, 2004.
15. D. Feldman, M. Monemizadeh, and C. Sohler. A PTAS for k-means clustering based on weak coresets. In SOCG '07: Proceedings of the twenty-third annual symposium on Computational geometry, pages 11-18, 2007. (Pubitemid 47582035)
16. B. Grünbaum. Venn diagrams and independent families of sets. Mathematics Magazine, 48(1):12-23, 1975.
17. S. Har-Peled. Coresets for Discrete Integration and Clustering. In proceedings of FSTTCS, pages 33-44, 2006.
18. S. Har-Peled and A. Kushal. Smaller coresets for k-median and k-means clustering. In SOCG '05: Proceedings of the twenty-first annual symposium on Computational geometry, pages 126-134, New York, NY, USA, 2005. ACM.
19. S. Har-Peled and S. Mazumdar. Coresets for k-means and k-median clustering and their applications. In Proc. 36th STOC, pages 291-300, 2004.
20. D. Haussler and E. Welzl. Epsilon-nets and simplex range queries. Discrete Comput Geom., 2:127-151, 1987.
21. D. Hilbert. Uber die Darstellung definiter Formen als Summe von Formenquadraten. Math. Ann., 32:342-350, 1888.
22. Ges Abh. 2:415-436, Chelsea Publishing Co., New York, 1965.
23. M. J. Kearns and R. E. Schapire. Efficient distribution-free learning of probabilistic concepts. J. Comput. System Sci., 48:464-497, 1994.
24. J. Kleinberg, C. Papadimitriou, and P. Raghavan. Approximation algorithms for segmentation problems. In Proc. 30th STOC, 1998.
25. R. Kress. Numerical Analysis. Springer, 1998.
26. M. Langberg and L. J. Schulman. Contraction and expansion of convex sets. In proceedings of CCCG, pages 25-28, 2004.
27. M. Langberg and L. J. Schulman. Universal ε-approximators for integrals. Manuscript, available at arXiv.org, 2009.
28. J. Matoušek. Lectures on Discrete Geometry. Springer, 2002.
29. D. Q. Naiman and H. P. Wynn. Independent Collections of Translates of Boxes and a Conjecture due to Grünbaum. Discrete and Computational Geometry, 9:101-105, 1993.
30. M. Naszódi and S. Taschuk. On the Transversal Number and VC-Dimension of Families of Positive Homothets of a Convex Body. Discrete Mathematics. To appear. Also http://arxiv.org/abs/0907.5223.
31. B. K. Natarajan. On learning sets and functions. Machine Learning, 4:67-97, 1989.
32. D. Pollard. Convergence of stochastic processes. Springer, 1984.
33. N. Sauer. On the density of families of sets. J. Combinatorial Theory Ser. A, 13:145-147, 1972.
34. L. J. Schulman. Clustering for edge-cost minimization. In Proc. 32nd STOC, pages 547-555, 2000.
35. S. Shelah. A combinatorial problem, stability and order for models and theories in infinitary languages. Pacific J. Math., 41:247-261, 1972.
36. V. N. Vapnik. Inductive principles of the search for empirical dependences. In Proc. 2nd Annual Workshop on Computational Learning Theory, 1989.
37. V. N. Vapnik and A. Y. Chervonenkis. On the uniform convergence of relative frequencies of events to their probabilities. Theory of Probability and its Applications, 16(2):264-280, 1971.
38. Wikipedia. Sparse grid. http://en.wikipedia.org/wiki/Sparse-grid. [Online; version of April 21, 2009].