A characterization of dimension free concentration in terms of transportation inequalities
By Nathael Gozlan. 2008
87 citations recorded by the source corpus.
The aim of this paper is to show that a probability measure concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand's $\T_2$ transportation-cost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincare inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9517082149721397fa848d9ebb3547f389d32932
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In Applied Mathematics
- Nathael Gozlan
- Cyril Roberto
- Prasad Tetali
- Paul-Marie Samson
- Patrick Cattiaux
- Arnaud Guillin
- Emanuel Milman
- Nicolas Juillet
In Mathematical Physics
Citations in the corpus
- Kantorovich duality for general transport costs and applications (2014, background): “In this section, extending [23], we characterize the transport-entropy inequality Tc(a1, a2) in terms of a dimension-free concentration property.”
- From the master equation to mean field game limit theory: Large deviations and concentration of measure (2018, unknown): “N. Gozlan and C. Léonard, Transport inequalities.”
- Characterization of a class of weak transport-entropy inequalities on the line (2015, unknown): “Conversely, it was shown by the first named author in [20] that a probability μ satisfying the dimension-free Gaussian concentration (3) necessarily satisfies T2(C), thus giving to Inequality T2 a special status among other functional inequalities appearing in the concentration of measure literature”
- Hamilton Jacobi equations on metric spaces and transport-entropy inequalities (2012, background): “The first proofs of the Otto–Villani theorem valid on any complete separable metric space appeared in [10] and [12].”