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Gualtierotti}, DOI = {10.1214/12-AIHP489}, URL = {https://doi.org/10.1214/12-AIHP489}, } @article {FoGu15, AUTHOR = {Fournier, N. and Guillin, A.}, TITLE = {On the rate of convergence in {W}asserstein distance of the empirical measure}, JOURNAL = {Probab. Theory Related Fields}, FJOURNAL = {Probability Theory and Related Fields}, VOLUME = {162}, YEAR = {2015}, NUMBER = {3-4}, PAGES = {707--738}, ISSN = {0178-8051}, MRCLASS = {60F25 (60E15 60F10)}, MRREVIEWER = {Jos\'e Trashorras}, DOI = {10.1007/s00440-014-0583-7}, URL = {https://doi.org/10.1007/s00440-014-0583-7}, } @article {BreLoe, AUTHOR = {Brenier, Y. and Loeper, G.}, TITLE = {A geometric approximation to the {E}uler equations: the {V}lasov-{M}onge-{A}mp\`ere system}, JOURNAL = {Geom. Funct. Anal.}, FJOURNAL = {Geometric and Functional Analysis}, VOLUME = {14}, YEAR = {2004}, NUMBER = {6}, PAGES = {1182--1218}, ISSN = {1016-443X}, MRCLASS = {35Q40 (35D05 76B99 76X05 82D10)}, MRREVIEWER = {Charles H.-T. 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The authors describe the interplay between the probabilistic structure (independence) and a variety of tools ranging from functional inequalities, transportation arguments, to information theory. Applications to the study of empirical processes, random projections, random matrix theory, and threshold phenomena are presented. The book offers a self-contained introduction to concentration inequalities, including a survey of concentration of sums of independent random variables, variance bounds, the entropy method, and the transportation method. Deep connections with isoperimetric problems are revealed. Special attention is paid to applications to the supremum of empirical processes.}, isbn = {9780199535255}, doi = {10.1093/acprof:oso/9780199535255.001.0001}, url = {https://doi.org/10.1093/acprof:oso/9780199535255.001.0001}, } @book{graf2000foundations, title={Foundations of quantization for probability distributions}, author={Graf, Siegfried and Luschgy, Harald}, year={2000}, publisher={Springer Science \& Business Media} } @article{carbone2023efficient, title={Efficient training of energy-based models using jarzynski equality}, author={Carbone, Davide and Hua, Mengjian and Coste, Simon and Vanden-Eijnden, Eric}, journal={Advances in Neural Information Processing Systems}, volume={36}, pages={52583--52614}, year={2023} } @Article{Ta94, Author = {M. {Talagrand}}, Title = {{The transportation cost from the uniform measure to the empirical measure in dimension $\geq 3$.}}, FJournal = {{The Annals of Probability}}, Journal = {{Ann. 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Scuola Normale Superiore di Pisa (Nuova Serie) }, VOLUME = {11}, EDITION = {Second}, PUBLISHER = {Edizioni della Normale, Pisa}, YEAR = {2012}, PAGES = {xiv+366}, ISBN = {978-88-7642-442-7; 978-88-7642-443-4}, DOI = {10.1007/978-88-7642-443-4}, URL = {http://dx.doi.org/10.1007/978-88-7642-443-4}, } @article{ambrosio2022quadratic, title={On the quadratic random matching problem in two-dimensional domains}, author={Ambrosio, Luigi and Goldman, Michael and Trevisan, Dario}, journal={Electronic Journal of Probability}, volume={27}, pages={1--35}, year={2022}, publisher={Institute of Mathematical Statistics and Bernoulli Society} } @article{caglioti2023random, title={Random matching in 2D with exponent 2 for densities defined on unbounded sets}, author={Caglioti, Emanuele and Pieroni, Francesca}, journal={arXiv preprint arXiv:2302.02602}, year={2023} } @article{goldman2021convergence, title={Convergence of asymptotic costs for random Euclidean matching problems}, author={Goldman, Michael and Trevisan, Dario}, journal={Probability and Mathematical Physics}, volume={2}, number={2}, pages={341--362}, year={2021}, publisher={Mathematical Sciences Publishers} } @misc{BobLe, title={A simple {F}ourier analytic proof of the {AKT} optimal matching theorem}, author={Sergey Bobkov and Michel Ledoux}, year={2019}, eprint={1909.06193}, archivePrefix={arXiv}, primaryClass={math.PR} } @book {VilOandN, AUTHOR = {Villani, C.}, TITLE = {Optimal transport: old and new}, SERIES = {Grundlehren der Mathematischen Wissenschaften }, VOLUME = {338}, PUBLISHER = {Springer-Verlag, Berlin}, YEAR = {2009}, PAGES = {xxii+973}, ISBN = {978-3-540-71049-3}, DOI = {10.1007/978-3-540-71050-9}, URL = {http://dx.doi.org/10.1007/978-3-540-71050-9}, } @article {CorderoMCannS, AUTHOR = {Cordero-Erausquin, D. and McCann, R. 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Mat., Nova S\'er.}}, ISSN = {0100-3569}, Volume = {32}, Number = {3}, Pages = {401--433}, Year = {2001}, Publisher = {Sociedade Brasileira de Matem\'atica, Rio de Janeiro; Springer, New York, NY}, MSC2010 = {28B05 35L65 35B35 76N10} } @article {ChenFried, AUTHOR = {Chen, G.-Q. and Frid, H.}, TITLE = {Extended divergence-measure fields and the {E}uler equations for gas dynamics}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {236}, YEAR = {2003}, NUMBER = {2}, PAGES = {251--280}, ISSN = {0010-3616}, MRCLASS = {35L65 (35Q35 76N10)}, MRREVIEWER = {Kenji Nishihara}, DOI = {10.1007/s00220-003-0823-7}, URL = {https://doi.org/10.1007/s00220-003-0823-7}, } @book {Viltop, AUTHOR = {Villani, C.}, TITLE = {Topics in optimal transportation}, SERIES = {Graduate Studies in Mathematics}, VOLUME = {58}, PUBLISHER = {American Mathematical Society, Providence, RI}, YEAR = {2003}, PAGES = {xvi+370}, ISBN = {0-8218-3312-X}, DOI = {10.1007/b12016}, URL = {http://dx.doi.org/10.1007/b12016}, } @book {AGS, AUTHOR = {Ambrosio, L. and Gigli, N. and Savar\'e, G.}, TITLE = {Gradient flows in metric spaces and in the space of probability measures}, SERIES = {Lectures in Mathematics ETH Z\"urich}, EDITION = {Second}, PUBLISHER = {Birkh\"auser Verlag, Basel}, YEAR = {2008}, PAGES = {x+334}, ISBN = {978-3-7643-8721-1}, } @book {Maggi, AUTHOR = {Maggi, F.}, TITLE = {Sets of finite perimeter and geometric variational problems}, SERIES = {Cambridge Studies in Advanced Mathematics}, VOLUME = {135}, NOTE = {An introduction to geometric measure theory}, PUBLISHER = {Cambridge University Press, Cambridge}, YEAR = {2012}, PAGES = {xx+454}, ISBN = {978-1-107-02103-7}, DOI = {10.1017/CBO9781139108133}, URL = {http://dx.doi.org/10.1017/CBO9781139108133}, } @book {Brezis, AUTHOR = {Br\'ezis, H.}, TITLE = {Op\'erateurs maximaux monotones et semi-groupes de contractions dans les espaces de {H}ilbert}, NOTE = {North-Holland Mathematics Studies, No. 5. 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