Transport Inequalities. A Survey

By Nathael Gozlan, Christian Léonard. 2010

178 citations recorded by the source corpus.

This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.

Subfields: Applied Mathematics, Transportation

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Citations in the corpus

Communication lower bounds for statistical estimation problems via a distributed data processing inequality (2015, background): “For a complete survey of transportation inequalities with other cost functions, please see the survey of Gozlan and L´eonard [GL10].”
Quantifying Distributional Model Risk via Optimal Transport (2016, unknown): “sporting mass between probability measures, and have been widely used in probability theory and its applications (see, for example, [43,44,53,2] for a massive collection of classical applications and [7,25,40,54,21,15,50,51,22] for a sample of growing list of new applications). Relative entropy (or)”
On a mixture of Brenier and Strassen Theorems (2018, background): “…back to the works of Marton [53, 52] and Talagrand [70, 71] in connections with the concentration of measure phenomenon via the so-called transport-entropy inequalities (see [48] for an introduction to concentration of measure phenomenon and [30] for a survey on transport-entropy inequalities).”
Log-Concavity and Strong Log-Concavity: a review (2014, background): “In their survey on optimal transport, Gozlan and Léonard (2010) show how to obtain the result of Bobkov and Ledoux from some transport inequalities.”
Kantorovich duality for general transport costs and applications (2014, background): “First we note, that when c ( x, p ) = ω ( x, y ) p ( dy ), (4.3) and (4.4) give back the usual transport-entropy inequalities of Talagrand type (see [37], [65] or [23] for a general introduction on the subject).”
A formula for the time derivative of the entropic cost and applications (2019, unknown): “. Then observe that log Z exp(ψ)dm= sup ˆZ ψdµ−H(µ|m) : µ∈ P(M), Z ψ+ dµ<∞ ˙ , where ψ+ := max{ψ,0}, the identity being a byproduct of the well-known variational representation of the entropy (see [19, 23]). Hence by taking the supremum over µin (A.4), we get the equivalent formulation Z QT 1 φdm”
On the mean speed of convergence of empirical and occupation measures in Wasserstein distance (2011, background): “For much more on the topic of transportation inequalities, the reader may refer to the survey [15] by N.”
Semi Log-Concave Markov Diffusions (2013, background): “For a nice survey on transportation inequalities we refer to [28].”
Simple Bounds for the Convergence of Empirical and Occupation Measures in 1-Wasserstein Distance (2011, background): “edian of f. Bobkov and Götze ([4]) were the ﬁrst to obtain an equivalent dual formulation of transportation inequalities. We present it here in a more general form obtained by Gozlan and Leonard (see [15]), in the case when the transportation cost function is the distance. Deﬁnition A.2. Let α: [0,+”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, background): “We also extend an argument of [5, 19] from the case of X = R1 to more general cases.”