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Entropy and the Combinatorial Dimension

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dc.contributor.authorMendelson, Shahar
dc.contributor.authorVershynin, Roman
dc.date.accessioned2015-12-13T23:07:12Z
dc.date.issued2003
dc.date.updated2015-12-12T08:08:17Z
dc.description.abstractWe 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 of consequences, including the optimal Elton's Theorem and estimates on the uniform central limit theorem in the real valued case.
dc.identifier.issn0020-9910
dc.identifier.urihttp://hdl.handle.net/1885/86106
dc.publisherSpringer
dc.sourceInventiones Mathematicae
dc.titleEntropy and the Combinatorial Dimension
dc.typeJournal article
local.bibliographicCitation.lastpage55
local.bibliographicCitation.startpage37
local.contributor.affiliationMendelson, Shahar, College of Engineering and Computer Science, ANU
local.contributor.affiliationVershynin, Roman, University of Alberta
local.contributor.authoruidMendelson, Shahar, u4011413
local.description.embargo2037-12-31
local.description.notesImported from ARIES
local.description.refereedYes
local.identifier.absfor010108 - Operator Algebras and Functional Analysis
local.identifier.ariespublicationMigratedxPub14856
local.identifier.citationvolume152
local.identifier.doi10.1007/s00222-002-0266-3
local.identifier.scopusID2-s2.0-0037596458
local.type.statusPublished Version

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