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Logarithmic Sobolev Inequalities and Concentration of Measure for Convex Functions and Polynomial Chaoses

100%
Adamczak R.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2005
|
tom 53
|
nr 2
221-238
2
Content available remote

Modified log-Sobolev inequalities for convex functions on the real line. Sufficient conditions

63%
Adamczak R., Strzelecki M.
Studia Mathematica
|
2015
|
tom 230
|
nr 1
59-93
EN
We provide a mild sufficient condition for a probability measure on the real line to satisfy a modified log-Sobolev inequality for convex functions, interpolating between the classical log-Sobolev inequality and a Bobkov-Ledoux type inequality. As a consequence we obtain dimension-free two-level concentration results for convex functions of independent random variables with sufficiently regular tail decay. We also provide a link between modified log-Sobolev inequalities for convex functions and weak transport-entropy inequalities, complementing recent work by Gozlan, Roberto, Samson, and Tetali.
3
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Chevet type inequality and norms of submatrices

45%
Adamczak R., Latała R., Litvak A., Pajor A., Tomczak-Jaegermann N.
Studia Mathematica
|
2012
|
tom 210
|
nr 1
35-56
EN
We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of "symmetric exponential" processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity $Γ_{k,m}$ that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates to give a sharp bound for the restricted isometry constant of a random matrix with independent log-concave unconditional rows. We also show that our Chevet type inequality does not extend to general isotropic log-concave random matrices.
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