Concentration of Measure Principle and Entropy-Inequalities

Samson, Paul-Marie

doi:10.1007/978-1-4939-7005-6_3

Cited by 9 publications

(10 citation statements)

References 95 publications

(123 reference statements)

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“…Remark The identity

{\bar{T}}_{2} false(ν false| μ false) = \underset{η \in B_{ν}}{trueprefixinf} W_{2}^{2} false(η, μ false)

of Proposition was already observed in [, Proposition 3.1] in dimension 1 when

μ

has no atom and then generalized to higher dimensions in [, Proposition 4.1] when

μ

is absolutely continuous with respect to Lebesgue. After a first version of this work has been released, we learned that Proposition has been independently obtained by Alfonsi, Corbetta and Jourdain in in connections with the study of algorithms approximating the martingale transport problem.…”

Section: Introductionmentioning

confidence: 87%

“…In particular, Kantorovich type duality formulas are obtained [33,Theorem 9.6] under the assumption that c is convex with respect to the p variable (and some additional mild regularity conditions). We refer to [22,31,32,66,68] for works directly connected to [33] and to [65] for an up-to-date survey of applications of weak transport costs to concentration of measure. Besides their many applications in the field of functional inequalities and concentration of measure, it turns out that weak transport costs are also interesting in themselves as a natural generalization of the transportation problem.…”

Section: More About Weak Optimal Transport Costsmentioning

confidence: 99%

“…In particular, Kantorovich type duality formulas are obtained [, Theorem 9.6] under the assumption that

c

is convex with respect to the

p

variable (and some additional mild regularity conditions). We refer to for works directly connected to and to for an up‐to‐date survey of applications of weak transport costs to concentration of measure.…”

Section: Introductionmentioning

confidence: 99%

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On a mixture of Brenier and Strassen Theorems

Gozlan

Juillet

2020

Proc. London Math. Soc.

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We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405). Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem (Caffarelli, Comm. Math. Phys. 214 (2000) 547–563).

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“…Remark The identity

{\bar{T}}_{2} false(ν false| μ false) = \underset{η \in B_{ν}}{trueprefixinf} W_{2}^{2} false(η, μ false)

of Proposition was already observed in [, Proposition 3.1] in dimension 1 when

μ

has no atom and then generalized to higher dimensions in [, Proposition 4.1] when

μ

Section: Introductionmentioning

confidence: 87%

Section: More About Weak Optimal Transport Costsmentioning

confidence: 99%

“…In particular, Kantorovich type duality formulas are obtained [, Theorem 9.6] under the assumption that

c

is convex with respect to the

p

Section: Introductionmentioning

confidence: 99%

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On a mixture of Brenier and Strassen Theorems

Gozlan

Juillet

2020

Proc. London Math. Soc.

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“…Above, the supremum runs over all measurable subset A of X. We refer to the survey [Sam16,Sam17] for other examples of weak transport-entropy inequalities and their connections with the concentration of measure principle.…”

Section: The Set Of Orbits Provides a Partition Of Gmentioning

confidence: 99%

Transport-entropy inequalities on locally acting groups of permutations

Samson¹

2017

Electron. J. Probab.

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Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G of permutations which act suitably 'locally'. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentration properties are consequences of these inequalities. The results are also generalised to a larger class of measures including Ewens distributions of arbitrary parameter θ on the symmetric group. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. These results are new examples, in discrete setting, of weak transport-entropy inequalities introduced in [GRST15], that contribute to a better understanding of the concentration properties of measures on permutations groups. One typical application is deviation bounds for the socalled configuration functions, such as the number of cycles of given lenght in the cycle decomposition of a random permutation.

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“…Another special case is the duality for backward convex order projection, this is previously proved by first establishing the equivalence between backward convex order projection and the weak optimal transport introduced in [25], and then using the duality theorem for the weak optimal transport. In one dimension, the equivalence is proved in [24], then generalized to higher dimensions [34] under the condition that µ has a density w.r.t. the Lebesgue measure.…”

mentioning

confidence: 99%

Backward and Forward Wasserstein Projections in Stochastic Order

Kim,

Ruan

2021

Preprint

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We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their characterizations require study on a case-by-case basis. Particular attention is given to convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. Our results extend to the forward convex order case the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to the early results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.

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Concentration of Measure Principle and Entropy-Inequalities

Cited by 9 publications

References 95 publications

On a mixture of Brenier and Strassen Theorems

On a mixture of Brenier and Strassen Theorems

Transport-entropy inequalities on locally acting groups of permutations

Backward and Forward Wasserstein Projections in Stochastic Order

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