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Entropy and the Combinatorial Dimension

Mendelson, Shahar; Vershynin, Roman


We solve Talagrand's entropy problem: The L2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley's theorem on classes of {0, l}-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...[Show more]

CollectionsANU Research Publications
Date published: 2003
Type: Journal article
Source: Inventiones Mathematicae
DOI: 10.1007/s00222-002-0266-3


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