Entropy and the Combinatorial Dimension

Details Entropy and the Combinatorial Dimension Entropy and the Combinatorial Dimension

2002-03-26

arxiv.org/abs/math/0203275v3

Mathematics

Functional Analysis

A final version of the paper (Inventiones Math., to appear) Only two applications added: one to Asymptotic Geometry (the optim

Scientific paper

10.1007/s00222-002-0266-3

We solve Talagrand's entropy problem: the L_2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case.

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