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Entropy and the Combinatorial Dimension

S. Mendelson·R. Vershynin
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Abstract

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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S. Mendelson, R. Vershynin. 2002-03-26. Entropy and the Combinatorial Dimension. https://arxiv.org/abs/math/0203275

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