The shattering dimension of sets of linear functionals

Details The shattering dimension of sets of linear functionals The shattering dimension of sets of linear functionals

2004-10-05

arxiv.org/abs/math/0410096v1

Annals of Probability 2004, Vol. 32, No. 3A, 1746-1770

Mathematics

Probability

Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imsta

Scientific paper

10.1214/009117904000000388

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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