Modified logarithmic Sobolev inequalities and transportation inequalities
By Ivan Gentil, Arnaud Guillin, Laurent Miclo. 2004
88 citations recorded by the source corpus.
Abstract.We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp (−|x|α) or exp (−|x|α log β(2+|x|)) (α ∈]1,2[ and β ∈ ℝ) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/c7faef04b013c51ef2490903d2f1ee14c99ec31b
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In Applied Mathematics
- Nathael Gozlan
- Arnaud Guillin
- Patrick Cattiaux
- Emanuel Milman
- Cyril Roberto
- Franck Barthe
- A. V. Kolesnikov
- Liming Wu
Citations in the corpus
- Lyapunov conditions for Super Poincaré inequalities (2009, background): “See [21, 6] for an attempt involving modified log-Sobolev inequalities introduced in [19] and mass transport.”
- Orlicz–Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups☆ (2006, background): “For other directions on the study of sub-Gaussian measures, the reader could like to see also [ GGM05 , BCR05a, BCR05c, Zeg01].”
- Characterization of Talagrand's like transportation-cost inequalities on the real line (2006, methodology): “[11] A. Guillin I. Gentil and L. Miclo.”
- Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry (2004, background): “For other results in connection with mass transportation, see also [59,23,29].”
- Poincaré inequalities and dimension free concentration of measure (2010, methodology): “As shown in [19] or [5], Example 31, if μ verifies LS(Hq,C) for some C then it verifies the concentration inequality (1.”
- Mass Transport and Variants of the Logarithmic Sobolev Inequality (2007, methodology): “Our contribution here is rather in the tightening techniques of Section 2 which yield a soft proof of the main results of [39, 40] with slightly relaxed conditions: Corollary 5.”
- Concentration inequalities for non-Lipschitz functions with bounded derivatives of higher order (2013, methodology): “3 in [28]), we can conclude the proof applying (70) to | f −E f | (similarly as in the proof of Theorem 3.”
- Applications of Stein's method for concentration inequalities (2009, background): “The papers [2, 14, 20, 28] deal with tails between exponential and Gaussian and [1, 4] deal with subexponential tails.”