The shattering dimension of sets of linear functionals

Annals of Probability

Volumn 32, Issue 3 A, 2004, Pages 1746-1770

  Mendelson, Shahar ^a   Schechtman, Gideon ^b  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

^b WEIZMANN INSTITUTE OF SCIENCE   (Israel)

Author keywords

Empirical processes; Linear functionals; Shattering dimension

Indexed keywords

EID: 4544289259     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000388     Document Type: Article

References (24)

1. ALON, N., BEN-DAVID, S., CESA-BIANCHI, N. and HAUSSLER, D. (1997). Scale sensitive dimensions, uniform convergence and learnability. J. Assoc. Comput. Mach. 44 615-631.
2. ANTHONY, M. and BARTLETT, P. L. (1999). Neural Network Learning, Theoretical Foundations. Cambridge Univ. Press.
3. BALL, K. (1989). Volumes of sections of cubes and related problems. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1376 251-260. Springer, Berlin.
4. BARTLETT, P. L., BOUSQUET, O. and MENDELSON, S. (2004). Localized Rademacher complexities. Ann. Statist. To appear.
5. BOUSQUET, O. (2002). A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Acad. Sci. Paris Ser. 1 334 495-500.
6. DUDLEY, R. M. (1999). Uniform Central Limit Theorems. Cambridge Univ. Press.
7. DUDLEY, R. M., GINÉ, E. and ZINN, J. (1991). Uniform and universal Glivenko-Cantelli classes. J. Theoret. Probab. 4 485-510.
8. GINÉ, E. and ZINN, J. (1991). Gaussian characterization of uniform Donsker classes of functions. Ann. Probab. 19 758-782.
9. GLUSKIN, E. D. (1978). Estimates of the norms of certain p-absolutely summing operators. Funktsional. Anal. i Prilozhen. 12 24-31. [Translated in Functional Anal. Appl. 12 94-101.]
10. GLUSKIN, E. D. (1978). Estimates of the norms of certain p-absolutely summing operators. Funktsional. Anal. i Prilozhen. 12 24-31. [Translated in Functional Anal. Appl. 12 94-101.]
11. GLUSKIN, E. D., PIETSCH, A. and PUHL, J. (1980). A generalization of Khintchine's inequality and its application in the theory of operator ideals. Studia Math. 67 149-155.
12. p spaces. Positivity 1 7-43.
13. p spaces. Studia Math. 136 147-182.
14. MENDELSON, S. (2002). Learnability in Hilbert spaces with reproducing kernels. J. Complexity 18 152-170.
15. MENDELSON, S. (2002). Rademacher averages and phase transitions in Glivenko-Cantelli class. IEEE Trans. Inform. Theory 48 251-263.
16. MENDELSON, S. (2003). A few notes on statistical learning theory. Advanced Lectures in Machine Learning. Lecture Notes in Comput. Sci. 2600 1-40. Springer, New York.
17. MENDELSON, S. and VERSHYNIN, R. (2003). Entropy and the combinatorial dimension. Invent. Math. 152 37-55.
18. n. J. Funct. Anal. 80 109-123.
19. MILMAN, V. D. and SCHECHTMAN, G. (1986). Asymptotic Theory of Finite Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin.
20. PISIER, G. (1989). The Volume of Convex Bodies and Banach Space Geometry. Cambridge Univ. Press.
21. TALAGRAND, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28-76.
22. TOMCZAK-JAEGERMANN, N. (1989). Banach-Mazur Distance and Finite-Dimensional Operator Ideals. CRC Press, Boca Raton, FL.
23. VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
24. VAPNIK, V. and CHERVONENKIS, A. (1971). Necessary and sufficient conditions for uniform convergence of means to mathematical expectations. Theory Probab. Appl. 26 532-553.