2020
DOI: 10.1007/s00526-020-01754-0
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A proof of the Caffarelli contraction theorem via entropic regularization

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Cited by 29 publications
(23 citation statements)
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“…Some of the following properties were already known for the softmax operator (for example in [39]) and they are used in order to get a posteriori regularity of the potentials but, up to our knowledge, were never used to get a priori results. Another very cleverly used properties of the (c, ε)-transform are in [32] in order to obtain a new proof of the Caffarelli's contraction theorem [12].…”
Section: Definition 22 (Entropic C-transform Ormentioning
confidence: 99%
See 1 more Smart Citation
“…Some of the following properties were already known for the softmax operator (for example in [39]) and they are used in order to get a posteriori regularity of the potentials but, up to our knowledge, were never used to get a priori results. Another very cleverly used properties of the (c, ε)-transform are in [32] in order to obtain a new proof of the Caffarelli's contraction theorem [12].…”
Section: Definition 22 (Entropic C-transform Ormentioning
confidence: 99%
“…[41,45]), optimal transport theory (e.g. [5,16,18,20,21,32,44,51,54,55,58]), data sciences (e.g. [34,39,40,56,57,62,63] see also the book [28] and references therein).…”
Section: Introductionmentioning
confidence: 99%
“…This gives the backward decomposition: given two probability measures, there is a transport plan between them given by the gradient map of a convex contraction, followed by a martingale coupling. This establishes a link with the celebrated Caffarelli's contraction theorem [15] (see also [30] and [18]) : if ν is a log-concave perturbation of the Gaussian measure µ, then the optimal transport map from µ to ν is given by the gradient of a convex function which is a contraction. In the language of item (2), the optimal map is a contraction when the projection μ to the backward cone is equal to ν.…”
mentioning
confidence: 70%
“…The existence of a connection between majorization and Caffarellli's contraction theorem seems to go back to Hargé [14] (see [12,11] for more recent results in this direction). Here we may put together our transport approach of the majorization in Lemma 2.4 and the latter theorem to get the following natural statement.…”
Section: 2mentioning
confidence: 93%