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July 2004 The shattering dimension of sets of linear functionals
Shahar Mendelson, Gideon Schechtman
Ann. Probab. 32(3): 1746-1770 (July 2004). DOI: 10.1214/009117904000000388

Abstract

We evaluate the shattering dimension of various classes of linear functionals on various symmetric convex sets. The proofs here relay mostly on methods from the local theory of normed spaces and include volume estimates, factorization techniques and tail estimates of norms, viewed as random variables on Euclidean spheres. The estimates of shattering dimensions can be applied to obtain error bounds for certain classes of functions, a fact which was the original motivation of this study. Although this can probably be done in a more traditional manner, we also use the approach presented here to determine whether several classes of linear functionals satisfy the uniform law of large numbers and the uniform central limit theorem.

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Shahar Mendelson. Gideon Schechtman. "The shattering dimension of sets of linear functionals." Ann. Probab. 32 (3) 1746 - 1770, July 2004. https://doi.org/10.1214/009117904000000388

Information

Published: July 2004
First available in Project Euclid: 14 July 2004

zbMATH: 1045.60006
MathSciNet: MR2073176
Digital Object Identifier: 10.1214/009117904000000388

Subjects:
Primary: 46B09 , 60D05

Keywords: Empirical processes , linear functionals , Shattering dimension

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 3 • July 2004
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