Entropy and the Combinatorial Dimension
S. Mendelson, R. Vershynin
Abstract
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 0,1-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.
Create a lesson
Related papers
Best proximity point results of operator type condensing mappings via measure of noncompactness and application to an infinite system of differential equations in c0 and 1
Parveen Kumar, Ankit Kumar
Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces
Konstantinos Konstantos, Pavlos Motakis, Thomas Speckhofer
The tensor product of the Fremlin tensor product of Banach lattice algebras
Jamel Jaber, Marwa Troudi
Gaussian inequalities with different covariance matrices - the Heterovariant Gaussian Brunn-Minkowski inequality
Emanuel Milman, Shohei Nakamura, Hiroshi Tsuji
Quasinormal and hyponormal operators in the β(H)-class
Ramesh Golla, Hiroyuki Osaka, Shanola S. Sequeira
A Vector-Valued Extension of a Theorem of Marciszewski
Leandro Candido