{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,25]],"date-time":"2026-07-25T03:13:34Z","timestamp":1784949214675,"version":"3.55.0"},"reference-count":438,"publisher":"Emerald","issue":"5-6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,2,12]]},"abstract":"<jats:p>Optimal transport (OT) theory can be informally described using the words of the French mathematician Gaspard Monge (1746-1818): A worker with a shovel in hand has to move a large pile of sand lying on a construction site. The goal of the worker is to erect with all that sand a target pile with a prescribed shape (for example, that of a giant sand castle). Naturally, the worker wishes to minimize her total effort, quantified for instance as the total distance or time spent carrying shovelfuls of sand. Mathematicians interested in OT cast that problem as that of comparing two probability distributions\u2014two different piles of sand of the same volume. They consider all of the many possible ways to morph, transport or reshape the first pile into the second, and associate a \u201cglobal\u201d cost to every such transport, using the \u201clocal\u201d consideration of how much it costs to move a grain of sand from one place to another. Mathematicians are interested in the properties of that least costly transport, as well as in its efficient computation. That smallest cost not only defines a distance between distributions, but it also entails a rich geometric structure on the space of probability distributions. That structure is canonical in the sense that it borrows key geometric properties of the underlying \u201cground\u201d space on which these distributions are defined. For instance, when the underlying space is Euclidean, key concepts such as interpolation, barycenters, convexity or gradients of functions extend naturally to the space of distributions endowed with an OT geometry.<\/jats:p>\n                  <jats:p>OT has been (re)discovered in many settings and under different forms, giving it a rich history. While Monge\u2019s seminal work was motivated by an engineering problem, Tolstoi in the 1920s and Hitchcock, Kantorovich and Koopmans in the 1940s established its significance to logistics and economics. Dantzig solved it numerically in 1949 within the framework of linear programming, giving OT a firm footing in optimization. OT was later revisited by analysts in the 1990s, notably Brenier, while also gaining fame in computer vision under the name of earth mover\u2019s distances. Recent years have witnessed yet another revolution in the spread of OT, thanks to the emergence of approximate solvers that can scale to large problem dimensions. As a consequence, OT is being increasingly used to unlock various problems in imaging sciences (such as color or texture processing), graphics (for shape manipulation) or machine learning (for regression, classification and generative modeling).<\/jats:p>\n                  <jats:p>This paper reviews OT with a bias toward numerical methods, and covers the theoretical properties of OT that can guide the design of new algorithms. We focus in particular on the recent wave of efficient algorithms that have helped OT find relevance in data sciences. We give a prominent place to the many generalizations of OT that have been proposed in but a few years, and connect them with related approaches originating from statistical inference, kernel methods and information theory. All of the figures can be reproduced using code made available in a companion website1 . This website hosts the book project Computational Optimal Transport. You will also find slides and computational resources.<\/jats:p>","DOI":"10.1561\/2200000073","type":"journal-article","created":{"date-parts":[[2019,2,12]],"date-time":"2019-02-12T05:27:49Z","timestamp":1549949269000},"page":"355-607","source":"Crossref","is-referenced-by-count":1928,"title":["Computational Optimal Transport with Applications to Data Sciences"],"prefix":"10.1108","volume":"11","author":[{"given":"Gabriel","family":"Peyr\u00e9","sequence":"first","affiliation":[{"name":"CNRS and DMA, ENS"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Marco","family":"Cuturi","sequence":"additional","affiliation":[{"name":"Google and CREST\/ENSAE"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"140","published-online":{"date-parts":[[2019,2,12]]},"reference":[{"key":"2026033012250439100_ref001","volume-title":"arXiv:1603.04467","author":"Abadi","year":"2016"},{"issue":"1","key":"2026033012250439100_ref002","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1007\/s00245-015-9323-3","article-title":"\u201cTomographic reconstruction from a few views: amulti-marginal optimal transportapproach\u201d","volume":"75","author":"Abraham","year":"2017","journal-title":"Applied Mathematics & Optimization"},{"key":"2026033012250439100_ref003","volume-title":"arXiv preprint arXiv:1106.1925","author":"Adams","year":"2011"},{"issue":"1","key":"2026033012250439100_ref004","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s10440-012-9783-2","article-title":"\u201cOne-dimensional numericalalgorithms for gradient flows in the p-Wassersteinspaces\u201d","volume":"125","author":"Agueh","year":"2013","journal-title":"Acta Applicandae Mathematicae"},{"issue":"2","key":"2026033012250439100_ref005","doi-asserted-by":"crossref","first-page":"904","DOI":"10.1137\/100805741","article-title":"\u201cBarycenters in the Wasserstein space\u201d","volume":"43","author":"Agueh","year":"2011","journal-title":"SIAM Journal on Mathematical Analysis"},{"issue":"7","key":"2026033012250439100_ref006","doi-asserted-by":"crossref","first-page":"812","DOI":"10.1016\/j.crma.2017.05.010","article-title":"\u201cVers un th\u00e9or\u00e8me dela limite centrale dans l\u2019espace deWasserstein?","volume":"355","author":"Agueh","year":"2017","journal-title":"Comptes Rendus Mathematique"},{"issue":"1","key":"2026033012250439100_ref007","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1111\/j.2517-6161.1966.tb00626.x","article-title":"\u201cA general class of coefficients ofdivergence of one distribution fromanother\u201d","volume":"28","author":"Ali","year":"1966","journal-title":"Journal of the Royal Statistical Society. Series B (Methodological)"},{"key":"2026033012250439100_ref008","volume-title":"arXiv:1704.02315","author":"Allen-Zhu","year":"2017"},{"key":"2026033012250439100_ref009","volume-title":"CoRR","author":"Al-Rfou","year":"2016"},{"key":"2026033012250439100_ref010","volume-title":"\u201cNear-linear time approximation algorithms for optimal transport via Sinkhorn iteration\u201d","author":"Altschuler","year":"2017"},{"issue":"2","key":"2026033012250439100_ref011","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1016\/j.jmaa.2016.04.045","article-title":"A fixed-point approach to barycenters in Wasserstein space","volume":"441","author":"#x00C4;lvarez-Esteban","year":"2016","journal-title":"Journal of Mathematical Analysis and Applications"},{"key":"2026033012250439100_ref012","author":"Alvarez-Melis","year":"2019","journal-title":"Towards Optimal Transport with Global Invariances"},{"issue":"1","key":"2026033012250439100_ref013","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1007\/s41884-018-0002-8","article-title":"Information geometry connecting Wasserstein distance and Kullback-Leibler divergence via the entropy-relaxed transportation problem","author":"Amari","year":"2018","journal-title":"Information Geometry"},{"key":"2026033012250439100_ref014","author":"Ambrosio","year":"2006"},{"issue":"2","key":"2026033012250439100_ref015","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1007\/s00205-008-0189-2","article-title":"Geodesics in the space of measurepreserving maps and plans","volume":"194","author":"Ambrosio","year":"2009","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"2","key":"2026033012250439100_ref016","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s00186-016-0549-x","article-title":"Discrete Wasserstein barycenters: optimal transport for discrete data","volume":"84","author":"Anderes","year":"2016","journal-title":"Mathematical Methods of Operations Research"},{"key":"2026033012250439100_ref017","first-page":"343","volume-title":"Society for Industrial and Applied Mathematics","author":"Andoni","year":"2008"},{"key":"2026033012250439100_ref018","doi-asserted-by":"crossref","first-page":"657","DOI":"10.24033\/asens.2363","article-title":"Snowflake universality of Wasserstein spaces","volume":"51","author":"Andoni","year":"2018","journal-title":"Annales scientifiques de l\u2019\u00c9cole normale sup\u00e9rieure"},{"key":"2026033012250439100_ref019","first-page":"214","article-title":"Wasserstein generative adversarial networks","volume":"70","author":"Arjovsky","year":"2017","journal-title":"Proceedings of the 34th International Conference on Machine Learning. Proceedings of Machine Learning Research"},{"issue":"1","key":"2026033012250439100_ref020","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1137\/0216006","article-title":"Power diagrams: properties, algorithms and applications","volume":"16","author":"Aurenhammer","year":"1987","journal-title":"SIAM Journal on Computing"},{"issue":"1","key":"2026033012250439100_ref021","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/PL00009187","article-title":"Minkowski-type theorems and least-squares clustering","volume":"20","author":"Aurenhammer","year":"1998","journal-title":"Algorithmica"},{"key":"2026033012250439100_ref022","volume-title":"Tech. Rep., Stanford University","author":"Averbuch","year":"2001"},{"key":"2026033012250439100_ref023","first-page":"384","article-title":"Self-concordant analysis for logistic regression","author":"Bach","year":"2010","journal-title":"Electronic Journal of Statistics"},{"issue":"1","key":"2026033012250439100_ref024","first-page":"595","article-title":"Adaptivity of averaged stochastic gradient descent to local strong convexity for logistic regression","volume":"15","author":"Bach","year":"2014","journal-title":"Journal of Machine Learning Research"},{"issue":"3","key":"2026033012250439100_ref025","doi-asserted-by":"crossref","first-page":"294","DOI":"10.2307\/2525582","article-title":"Estimating nonnegative matrices from marginal data","volume":"6","author":"Bacharach","year":"1965","journal-title":"International Economic Review"},{"key":"2026033012250439100_ref026","author":"Barrio","year":"2016"},{"key":"2026033012250439100_ref027","volume-title":"Graduate Studies in Mathematics","author":"Barvinok","year":"2002"},{"issue":"12","key":"2026033012250439100_ref028","doi-asserted-by":"crossref","first-page":"1298","DOI":"10.1016\/j.spl.2006.02.001","article-title":"On minimum Kantorovich distance estimators","volume":"76","author":"Bassetti","year":"2006","journal-title":"Statistics & Probability Letters"},{"key":"2026033012250439100_ref029","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-9467-7","author":"Bauschke","year":"2011","journal-title":"Convex analysis and monotone operator theory in Hilbert spaces"},{"issue":"4","key":"2026033012250439100_ref030","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1080\/02331930008844513","article-title":"Dykstra\u2019s algorithm with Bregman projections: a convergence proof","volume":"48","author":"Bauschke","year":"2000","journal-title":"Optimization"},{"issue":"3","key":"2026033012250439100_ref031","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0167-6377(02)00231-6","article-title":"Mirror descent and nonlinear projected subgradient methods for convex optimization","volume":"31","author":"Beck","year":"2003","journal-title":"Operations Research Letters"},{"key":"2026033012250439100_ref032","doi-asserted-by":"crossref","first-page":"643","DOI":"10.2307\/1907646","article-title":"A continuous model of transportation","volume":"20","author":"Beckmann","year":"1952","journal-title":"Econometrica"},{"issue":"3","key":"2026033012250439100_ref033","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s00780-013-0205-8","article-title":"Modelindependent bounds for option prices: a mass transport approach","volume":"17","author":"Beiglbock","journal-title":"Finance and Stochastics"},{"issue":"1","key":"2026033012250439100_ref034","first-page":"17","article-title":"Nonparametric entropy estimation: an overview","volume":"6","author":"Beirlant","year":"1997","journal-title":"International Journal of Mathematical and Statistical Sciences"},{"issue":"05","key":"2026033012250439100_ref035","doi-asserted-by":"crossref","first-page":"851","DOI":"10.1051\/m2an:2003058","article-title":"Numerical resolution of an \u201cunbalanced\u201d mass transport problem","volume":"37","author":"Benamou","year":"2003","journal-title":"ESAIM: Mathematical Model ling and Numerical Analysis"},{"issue":"3","key":"2026033012250439100_ref036","doi-asserted-by":"crossref","DOI":"10.1007\/s002110050002","article-title":"A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem","volume":"84","author":"Benamou","year":"2000","journal-title":"Numerische Mathematik"},{"issue":"1","key":"2026033012250439100_ref037","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10957-015-0725-9","article-title":"Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations","volume":"167","author":"Benamou","year":"2015","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"2","key":"2026033012250439100_ref038","doi-asserted-by":"crossref","first-page":"A1111","DOI":"10.1137\/141000439","article-title":"Iterative Bregman projections for regularized transportation problems","volume":"37","author":"Benamou","year":"2015","journal-title":"SIAM Journal on Scientific Computing"},{"issue":"3","key":"2026033012250439100_ref039","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1007\/s00211-015-0781-y","article-title":"Discretization of functionals involving the Monge-Ampere operator","volume":"134","author":"Benamou","year":"2016","journal-title":"Numerische Mathematik"},{"issue":"302","key":"2026033012250439100_ref040","doi-asserted-by":"crossref","first-page":"2743","DOI":"10.1090\/mcom\/3080","article-title":"Monotone and consistent discretization of the Monge-Ampere operator","volume":"85","author":"Benamou","year":"2016","journal-title":"Mathematics of Computation"},{"key":"2026033012250439100_ref041","volume-title":"Journal of Computational Physics","author":"Benamou","year":"2014"},{"key":"2026033012250439100_ref042","first-page":"No.100","volume-title":"Graduate Texts in Mathematics","author":"Berg","year":"1984"},{"key":"2026033012250439100_ref043","volume-title":"Reproducing Kernel Hilbert Spaces in Probability and Statistics","author":"Berlinet","year":"2003"},{"key":"2026033012250439100_ref044","first-page":"1777","volume-title":"Proceedings of the 31st Conference On Learning Theory","author":"Bernton","year":"2018"},{"key":"2026033012250439100_ref045","volume-title":"arXiv preprint arXiv:1701.05146","author":"Bernton","year":"2017"},{"issue":"1","key":"2026033012250439100_ref046","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1007\/BF01584237","volume":"21","author":"Bertsekas","year":"1981","journal-title":"Mathematical Programming"},{"issue":"1","key":"2026033012250439100_ref047","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF00247653","article-title":"Auction algorithms for network flow problems: a tutorial introduction","volume":"1","author":"Bertsekas","year":"1992","journal-title":"Computational Optimization and Applications"},{"key":"2026033012250439100_ref048","volume-title":"Network Optimization: Continuous and Discrete Models Athena Scientific","author":"Bertsekas","year":"1998"},{"issue":"1","key":"2026033012250439100_ref049","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/BF01589405","article-title":"Dual coordinate step methods for linear network flow problems","volume":"42","author":"Bertsekas","year":"1988","journal-title":"Mathematical Programming"},{"key":"2026033012250439100_ref050","volume-title":"Introduction to Linear Optimization","author":"Bertsimas","year":"1997"},{"key":"2026033012250439100_ref051","author":"Bhatia","year":"2018","journal-title":"Expositiones Mathe- maticae, to appear"},{"key":"2026033012250439100_ref052","volume-title":"arXiv preprint arXiv:1711.08947","author":"Bigot","year":"2017"},{"issue":"1","key":"2026033012250439100_ref053","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/15-AIHP706","article-title":"Geodesic PCA in the Wasserstein space by convex PCA","volume":"53","author":"Bigot","year":"2017","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques"},{"key":"2026033012250439100_ref054","volume-title":"arXiv:1212.2562","author":"Bigot","year":"2012"},{"key":"2026033012250439100_ref055","author":"Bigot","year":"2012","journal-title":"\u201cConsistent estimation of a population barycenter in the Wasserstein space\u201d"},{"key":"2026033012250439100_ref056","first-page":"147","article-title":"Tres observaciones sobre el algebra lineal","volume":"5","author":"Birkhoff","year":"1946","journal-title":"Universidad Nacional de Tucum\u00e1n Revista Series A"},{"issue":"1","key":"2026033012250439100_ref057","first-page":"219","volume":"85","author":"Birkhoff","year":"1957","journal-title":"Transactions of the American Mathematical Society"},{"issue":"2","key":"2026033012250439100_ref058","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1137\/070683337","article-title":"Convergence of the mass-transport steepest descent scheme for the subcritical Patlak- Keller-Segel model","volume":"46","author":"Blanchet","year":"2008","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"1","key":"2026033012250439100_ref059","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1287\/moor.2015.0719","article-title":"Optimal transport and Cournot- Nash equilibria","volume":"41","author":"Blanchet","year":"2015","journal-title":"Mathematics of Operations Research"},{"key":"2026033012250439100_ref060","first-page":"16: 2296","volume-title":"Electronic Journal of Probability","author":"Boissard","year":"2011"},{"issue":"2","key":"2026033012250439100_ref061","article-title":"Distribution\u2019s template estimate with Wasserstein metrics","volume":"21","author":"Boissard","year":"2015","journal-title":"Bernoul li"},{"issue":"3","key":"2026033012250439100_ref062","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1007\/s00440-006-0004-7","article-title":"Quantitative concentration inequalities for empirical measures on non-compact spaces","volume":"137","author":"Bolley","year":"2007","journal-title":"Probability Theory and Related Fields"},{"issue":"4","key":"2026033012250439100_ref063","doi-asserted-by":"crossref","first-page":"71:1","DOI":"10.1145\/2897824.2925918","article-title":"Wasserstein barycentric coordinates: histogram regression using optimal transport","volume":"35","author":"Bonneel","year":"2016","journal-title":"ACM Transactions on Graphics"},{"issue":"1","key":"2026033012250439100_ref064","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1007\/s10851-014-0506-3","article-title":"Sliced and Radon Wasserstein barycenters of measures","volume":"51","author":"Bonneel","year":"2015","journal-title":"Journal of Mathematical Imaging and Vision"},{"issue":"6","key":"2026033012250439100_ref065","doi-asserted-by":"crossref","first-page":"No.158","DOI":"10.1145\/2070781.2024192","article-title":"Displacement interpolation using Lagrangian mass transport","volume":"30","author":"Bonneel","year":"2011","journal-title":"ACM Transactions on Graphics"},{"key":"2026033012250439100_ref066","first-page":"297","article-title":"De investigando ordine systematis aequationum differentialium vulgarium cujuscunque","volume":"64","author":"Borchardt","year":"1865","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"key":"2026033012250439100_ref067","volume-title":"Modern Multidimensional Scaling: Theory and Applications","author":"Borg","year":"2005"},{"key":"2026033012250439100_ref068","author":"Botsch","year":"2010"},{"key":"2026033012250439100_ref069","volume-title":"\u201cFrom optimal transport to generative modeling: the VEGAN cookbook\u201d","author":"Bousquet","year":"2017"},{"issue":"1","key":"2026033012250439100_ref070","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1561\/2200000016","article-title":"\u201cDistributed optimization and statistical learning via the alternating direction method of multipliers\u201d","volume":"3","author":"Boyd","year":"2011","journal-title":"Foundations and Trends in Machine Learning"},{"issue":"3","key":"2026033012250439100_ref0440","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","article-title":"\u201cThe relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming\u201d","volume":"7","author":"Bregman","year":"1967","journal-title":"USSR Computational Mathematics and Mathematical Physics"},{"issue":"19","key":"2026033012250439100_ref071","first-page":"805","volume":"305","author":"Brenier","year":"1987","journal-title":"\u201cD\u00e9composition polaire et r\u00e9arrangement monotone des champs de vecteurs\u201d"},{"key":"2026033012250439100_ref072","first-page":"225","article-title":"\u201cThe least action principle and the related concept of generalized flows for incompressible perfect fluids\u201d","volume":"2","author":"Brenier","year":"1990","journal-title":"Journal of the AMS"},{"issue":"4","key":"2026033012250439100_ref073","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1002\/cpa.3160440402","article-title":"\u201cPolar factorization and monotone rearrangement of vector-valued functions\u201d","volume":"44","author":"Brenier","year":"1991","journal-title":"Communications on Pure and Applied Mathematics"},{"issue":"4","key":"2026033012250439100_ref074","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/BF00375139","article-title":"\u201cThe dual least action problem for an ideal, incompressible fluid\u201d","volume":"122","author":"Brenier","year":"1993","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"4","key":"2026033012250439100_ref075","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1002\/(SICI)1097-0312(199904)52:4<411::AID-CPA1>3.0.CO;2-3","article-title":"\u201cMinimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations\u201d","volume":"52","author":"Brenier","year":"1999","journal-title":"Communications on Pure and Applied Mathematics"},{"issue":"1417","key":"2026033012250439100_ref076","doi-asserted-by":"crossref","first-page":"1982","DOI":"10.1016\/j.physd.2008.02.026","article-title":"\u201cGeneralized solutions and hydrostatic approximation of the Euler equations\u201d","volume":"237","author":"Brenier","year":"2008","journal-title":"Physica D. Nonlinear Phenomena"},{"issue":"5","key":"2026033012250439100_ref077","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1073\/pnas.0508601103","article-title":"\u201cGeneralized multidimensional scaling: a framework for isometry-invariant partial surface matching\u201d","volume":"103","author":"Bronstein","year":"2006","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"2-3","key":"2026033012250439100_ref078","doi-asserted-by":"crossref","first-page":"266","DOI":"10.1007\/s11263-009-0301-6","article-title":"\u201cA Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching\u201d","volume":"89","author":"Bronstein","year":"2010","journal-title":"International Journal on Computer Vision"},{"key":"2026033012250439100_ref079","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511721182","volume":"Vol. 108","author":"Brualdi","year":"2006","journal-title":"Combinatorial Matrix Classes"},{"key":"2026033012250439100_ref080","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/033","volume":"Vol. 33","author":"Burago","year":"2001","journal-title":"A Course in Metric Geometry"},{"key":"2026033012250439100_ref081","first-page":"199","article-title":"\u201cAn extension of Kakutani\u2019s theorem on infinite product measures to the tensor product of semifinite w*-algebras\u201d","volume":"135","author":"Bures","year":"1969","journal-title":"Transactions of the American Mathematical Society"},{"issue":"1","key":"2026033012250439100_ref082","doi-asserted-by":"crossref","first-page":"59","DOI":"10.3934\/krm.2010.3.59","article-title":"\u201cA mixed finite element method for nonlinear diffusion equations\u201d","volume":"3","author":"Burger","year":"2010","journal-title":"Kinetic and Related Models"},{"key":"2026033012250439100_ref083","first-page":"209","volume":"2","author":"Burger","year":"2012","journal-title":"\u201cRegularised regression and density estimation based on optimal transport\u201d"},{"issue":"6","key":"2026033012250439100_ref084","first-page":"062","volume":"85","author":"Buttazzo","year":"2012","journal-title":"\u201cOptimaltransport formulation of electronic density-functional theory\u201d"},{"key":"2026033012250439100_ref085","first-page":"1","volume-title":"\u201cThe Monge-Ampere equation and optimal transportation, an elementary review","author":"Caffarelli","year":"2003"},{"issue":"13","key":"2026033012250439100_ref086","article-title":"\u201cProblem of reflector design with given far-field scattering data\u201d","volume":"Vol. 226","author":"Caffarelli","year":"1999","journal-title":"Monge Amp\u00e8re equation: applications to geometry and optimization"},{"issue":"2","key":"2026033012250439100_ref087","doi-asserted-by":"crossref","first-page":"673","DOI":"10.4007\/annals.2010.171.673","article-title":"#x201C;Free boundaries in optimal transport and Monge-Amp\u00e8re obstacle problems\u201d","volume":"171","author":"Caffarelli","year":"2010","journal-title":"Annals of Mathematics"},{"issue":"1","key":"2026033012250439100_ref088","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1090\/S0894-0347-01-00376-9","article-title":"\u201cConstructing optimal maps for Monge\u2019s transport problem as a limit of strictly convex costs\u201d","volume":"15","author":"Caffarelli","year":"2002","journal-title":"Journal of the American Mathematical Society"},{"key":"2026033012250439100_ref089","first-page":"2492","volume-title":"Advances in Neural Information Processing Systems 25","author":"Canas","year":"2012"},{"issue":"3","key":"2026033012250439100_ref090","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1007\/s00220-014-2124-8","article-title":"\u201cAn analog of the 2-Wasserstein metric in non-commutative probability under which the fermionic Fokker- Planck equation is gradient flow for the entropy\u201d","volume":"331","author":"Carlen","year":"2014","journal-title":"Communications in Mathematical Physics"},{"key":"2026033012250439100_ref091","volume-title":"\u201cVector quantile regression beyond correct specification\u201d","author":"Carlier","year":"2016"},{"issue":"2","key":"2026033012250439100_ref092","first-page":"1385","volume":"49","author":"Carlier","year":"2017","journal-title":"\u201cConvergence of entropic schemes for optimal transport and gradient flows\u201d"},{"issue":"2","key":"2026033012250439100_ref093","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/s00199-008-0415-z","article-title":"\u201cMatching for teams\u201d","volume":"42","author":"Carlier","year":"2010","journal-title":"Economic Theory"},{"issue":"6","key":"2026033012250439100_ref094","doi-asserted-by":"crossref","first-page":"2554","DOI":"10.1137\/080740647","article-title":"\u201cFrom Knothe\u2019s transport to Brenier\u2019s map and a continuation method for optimal transport\u201d","volume":"41","author":"Carlier","year":"2010","journal-title":"SIAM Journal on Mathematical Analysis"},{"issue":"3","key":"2026033012250439100_ref095","doi-asserted-by":"crossref","first-page":"1330","DOI":"10.1137\/060672832","article-title":"\u201cOptimal transportation with traffic congestion and Wardrop equilibria\u201d","volume":"47","author":"Carlier","year":"2008","journal-title":"SIAM Journal on Control and Optimization"},{"issue":"6","key":"2026033012250439100_ref096","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1051\/m2an\/2015033","article-title":"\u201cNumerical methods for matching for teams and Wasserstein barycenters\u201d","volume":"49","author":"Carlier","year":"2015","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"2026033012250439100_ref097","volume-title":"to appear in ESAIM: COCV","author":"Carlier","year":"2019"},{"issue":"01","key":"2026033012250439100_ref098","doi-asserted-by":"crossref","first-page":"233","DOI":"10.4208\/cicp.160214.010814a","article-title":"\u201cA finite-volume method for nonlinear nonlocal equations with a gradient flow structure\u201d","volume":"17","author":"Carrillo","year":"2015","journal-title":"Communications in Computational Physics"},{"issue":"6","key":"2026033012250439100_ref099","doi-asserted-by":"crossref","first-page":"4305","DOI":"10.1137\/080739574","article-title":"\u201cNumerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms\u201d","volume":"31","author":"Carrillo","year":"2009","journal-title":"SIAM Journal on Scientific Computing"},{"key":"2026033012250439100_ref100","first-page":"407","article-title":"\u201cThe Dykstra algorithm with Bregman projections\u201d","volume":"2","author":"Censor","year":"1998","journal-title":"Communications in Applied Analysis"},{"issue":"3","key":"2026033012250439100_ref101","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1007\/BF00940051","article-title":"\u201cProximal minimization algorithm with-functions\u201d","volume":"73","author":"Censor","year":"1992","journal-title":"Journal of Optimization Theory and Applications"},{"issue":"1","key":"2026033012250439100_ref102","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/07069938X","article-title":"\u201cThe ^-Wasserstein distance: local solutions and existence of optimal transport maps\u201d","volume":"40","author":"Champion","year":"2008","journal-title":"SIAM Journal on Mathematical Analysis"},{"issue":"4","key":"2026033012250439100_ref103","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1007\/BF02712873","article-title":"\u201cOptimal output-sensitive convex hull algorithms in two and three dimensions\u201d","volume":"16","author":"Chan","year":"1996","journal-title":"Discrete & Computational Geometry"},{"key":"2026033012250439100_ref104","volume-title":"\u201cOn the matrix Monge-Kantorovich problem\u201d","author":"Chen","year":"2017"},{"issue":"2","key":"2026033012250439100_ref105","doi-asserted-by":"crossref","first-page":"671","DOI":"10.1007\/s10957-015-0803-z","article-title":"\u201cOn the relation between optimal transport and Schrodinger bridges: A stochastic control viewpoint\u201d","volume":"169","author":"Chen","year":"2016","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026033012250439100_ref106","volume-title":"\u201cMatrix optimal mass transport: a quantum mechanical approach\u201d","author":"Chen","year":"2016"},{"issue":"1","key":"2026033012250439100_ref107","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10208-016-9331-y","article-title":"\u201cAn interpolating distance between optimal transport and Fisher-Rao metrics\u201d","volume":"18","author":"Chizat","year":"2018","journal-title":"Foundations of Computational Mathematics"},{"key":"2026033012250439100_ref108","doi-asserted-by":"crossref","first-page":"2563","DOI":"10.1090\/mcom\/3303","article-title":"\u201cScaling algorithms for unbalanced transport problems\u201d","volume":"87","author":"Chizat","year":"2018","journal-title":"Mathematics of Computation"},{"issue":"11","key":"2026033012250439100_ref109","doi-asserted-by":"crossref","first-page":"3090","DOI":"10.1016\/j.jfa.2018.03.008","article-title":"\u201cUnbalanced optimal transport: geometry and Kantorovich formulation\u201d","volume":"274","author":"Chizat","year":"2018","journal-title":"Journal of Functional Analysis"},{"issue":"3","key":"2026033012250439100_ref110","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s00205-011-0471-6","article-title":"\u201cFokker-Planck equations for a free energy functional or Markov process on a graph\u201d","volume":"203","author":"Chow","year":"2012","journal-title":"Archive for Rational Mechanics and Analysis"},{"key":"2026033012250439100_ref111","volume-title":"\u201cA discrete Schrodinger equation via optimal transport on graphs\u201d","author":"Chow","year":"2017"},{"key":"2026033012250439100_ref112","volume-title":"\u201cEntropy dissipation of Fokker- Planck equations on graphs","author":"Chow","year":"2017"},{"key":"2026033012250439100_ref113","volume-title":"\u201cConstructing geodesics on the space of compact metric spaces\u201d","author":"Chowdhury","year":"2016"},{"key":"2026033012250439100_ref114","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1109\/CVPR.2000.854733","article-title":"\u201cA new algorithm for non-rigid point matching\u201d","volume":"Vol. 2","author":"Chui","year":"2000","journal-title":"Computer Vision and Pattern Recognition, 2000. Proceedings. IEEE Conference on"},{"key":"2026033012250439100_ref115","first-page":"299","article-title":"\u201cInformation-type measures of difference of probability distributions and indirect observations\u201d","volume":"2","author":"Ciszar","year":"1967","journal-title":"Studia Scientiarum Mathematicarum Hungarica"},{"key":"2026033012250439100_ref116","doi-asserted-by":"crossref","DOI":"10.1109\/FOCS.2017.88","volume-title":"\u201cMatrix scaling and balancing via box constrained Newton\u2019s method and interior point methods\u201d","author":"Cohen","year":"2017"},{"key":"2026033012250439100_ref117","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1109\/ICCV.1999.790393","article-title":"\u201cThe earth mover\u2019s distance under transformation sets\u201d","volume":"Vol. 2","author":"Cohen","year":"1999","journal-title":"Proceedings of the Seventh IEEE International Conference on Computer vision"},{"issue":"4","key":"2026033012250439100_ref118","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1109\/JSTSP.2007.910264","article-title":"\u201cA Douglas-Rachford splitting approach to nonsmooth convex variational signal recovery\u201d","volume":"1","author":"Combettes","year":"2007","journal-title":"IEEE Journal of Selected Topics in Signal Processing"},{"issue":"1-3","key":"2026033012250439100_ref119","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF01582220","article-title":"\u201cAsymptotic analysis of the exponential penalty trajectory in linear programming\u201d","volume":"67","author":"Cominetti","year":"1994","journal-title":"Mathematical Programming"},{"key":"2026033012250439100_ref120","first-page":"1","volume-title":"\u201cFast projection onto the simplex and the i 1 ball\u201d","author":"Condat","year":"2015"},{"key":"2026033012250439100_ref121","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.dam.2014.10.004","article-title":"\u201cFisher information distance: a geometrical reading\u201d","volume":"197","author":"Costa","year":"2015","journal-title":"Discrete Applied Mathematics"},{"issue":"4","key":"2026033012250439100_ref122","doi-asserted-by":"crossref","first-page":"548","DOI":"10.1002\/cpa.21437","article-title":"\u201cDensity functional theory and optimal transportation with Coulomb cost\u201d","volume":"66","author":"Cotar","year":"2013","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"2026033012250439100_ref123","first-page":"3730","volume-title":"Advances in Neural Information Processing Systems 30","author":"Courty","year":"2017"},{"key":"2026033012250439100_ref124","first-page":"3571","volume-title":"2016 IEEE International Geoscience and Remote Sensing Symposium","author":"Courty","year":"2016"},{"issue":"9","key":"2026033012250439100_ref125","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1109\/TPAMI.2016.2615921","article-title":"\u201cOptimal transport for domain adaptation\u201d","volume":"39","author":"Courty","year":"2017","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"issue":"5","key":"2026033012250439100_ref126","doi-asserted-by":"crossref","first-page":"152:1","DOI":"10.1145\/2516971.2516977","article-title":"\u201cGeodesics in heat: a new approach to computing distance based on heat flow\u201d","volume":"32","author":"Crane","year":"2013","journal-title":"ACM Transaction on Graphics"},{"issue":"3","key":"2026033012250439100_ref127","doi-asserted-by":"crossref","first-page":"1264","DOI":"10.1214\/aop\/1176991269","article-title":"\u201cNotes on the Wasserstein Metric in Hilbert Spaces\u201d","volume":"17","author":"Cuesta","year":"1989","journal-title":"The Annals of Probability"},{"key":"2026033012250439100_ref128","volume-title":"\u201cPositivity and transportation\u201d","author":"Cuturi","year":"2012"},{"key":"2026033012250439100_ref129","first-page":"2292","volume-title":"Advances in Neural Information Processing Systems 26","author":"Cuturi","year":"2013"},{"key":"2026033012250439100_ref130","first-page":"533","article-title":"#x201C;Ground metric learning\u201d","volume":"15","author":"Cuturi","year":"2014","journal-title":"Journal of Machine Learning Research"},{"key":"2026033012250439100_ref131","first-page":"685","article-title":"\u201cFast computation of Wasserstein barycenters\u201d","volume":"Vol. 32.","author":"Cuturi","year":"2014","journal-title":"Proceedings of ICML"},{"key":"2026033012250439100_ref132","doi-asserted-by":"crossref","first-page":"329","DOI":"10.7551\/mitpress\/7503.003.0046","volume-title":"Advances in Neural Information Processing Systems 19","author":"Cuturi","year":"2007"},{"issue":"1","key":"2026033012250439100_ref133","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1137\/15M1032600","article-title":"\u201cA smoothed dual approach for variational Wasserstein problems\u201d","volume":"9","author":"Cuturi","year":"2016","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"4","key":"2026033012250439100_ref134","first-page":"941","volume":"60","author":"Cuturi","year":"2018","journal-title":"\u201cSemidual Regularized Optimal Transport\u201d"},{"key":"2026033012250439100_ref135","first-page":"678","article-title":"\u201cFurther and stronger analogy between sampling and optimization: Langevin Monte Carlo and gradient descent\u201d","author":"Dalalyan","year":"2017","journal-title":"Proceedings of the 2017 Conference on Learning Theory. Vol. 65. Proceedings of Machine Learning Research"},{"key":"2026033012250439100_ref136","volume-title":"\u201cUser-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient\u201d","author":"Dalalyan","year":"2017"},{"issue":"3\/4","key":"2026033012250439100_ref137","doi-asserted-by":"crossref","first-page":"200","DOI":"10.2307\/1905523","article-title":"\u201cProgramming of interdependent activities: II mathematical model\u201d","volume":"17","author":"Dantzig","year":"1949","journal-title":"Econometrica"},{"key":"2026033012250439100_ref138","first-page":"359","article-title":"\u201cApplication of the simplex method to a transportation problem\u201d","volume":"13","author":"Dantzig","year":"1951","journal-title":"Activity Analysis of Production and Allocation"},{"key":"2026033012250439100_ref139","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1007\/978-3-642-68874-4_4","volume-title":"Mathematical Programming The State of the Art: Bonn 1982","author":"Dantzig","year":"1983"},{"key":"2026033012250439100_ref140","first-page":"257","volume-title":"History of mathematical programming: a col lection of personal reminiscences","author":"Dantzig","year":"1991"},{"key":"2026033012250439100_ref141","volume-title":"Convex Optimization & Euclidean Distance Geometry","author":"Dattorro","year":"2017"},{"issue":"6","key":"2026033012250439100_ref142","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1145\/2366145.2366190","article-title":"\u201cBlue noise through optimal transport\u201d","volume":"31","author":"De Goes","year":"2012","journal-title":"ACM Transactions on Graphics"},{"issue":"2","key":"2026033012250439100_ref143","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1023\/B:JMIV.0000035178.72139.2d","article-title":"\u201cMidway image equalization\u201d","volume":"21","author":"Delon","year":"2004","journal-title":"Journal of Mathematical Imaging and Vision"},{"issue":"7","key":"2026033012250439100_ref144","doi-asserted-by":"crossref","first-page":"2239","DOI":"10.1137\/090772708","article-title":"\u201cFast transport optimization for Monge costs on the circle\u201d","volume":"70","author":"Delon","year":"2010","journal-title":"SIAM Journal on Applied Mathematics"},{"issue":"l2","key":"2026033012250439100_ref145","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1137\/110823304","article-title":"\u201cLocal matching indicators for transport problems with concave costs\u201d","volume":"26","author":"Delon","year":"2012","journal-title":"SIAM Journal on Discrete Mathematics"},{"issue":"4","key":"2026033012250439100_ref146","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1214\/aoms\/1177731829","article-title":"\u201cOn a least squares adjustment of a sampled frequency table when the expected marginal totals are known\u201d","volume":"11","author":"Deming","year":"1940","journal-title":"Annals of Mathematical Statistics"},{"issue":"4","key":"2026033012250439100_ref147","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1214\/12-AIHP489","article-title":"\u201cConstructive quantization: Approximation by empirical measures\u201d","volume":"49.","author":"Dereich","year":"2013","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques"},{"key":"2026033012250439100_ref148","volume-title":"\u201cRecursively implementating the Gaussian and its derivatives\u201d","author":"Deriche","year":"1993"},{"key":"2026033012250439100_ref149","author":"Dessein","year":"2017"},{"issue":"15","key":"2026033012250439100_ref150","first-page":"1","article-title":"Regularized Optimal Transport and the Rot Mover","volume":"19","author":"Dessein","year":"2018"},{"key":"2026033012250439100_ref151","author":"Di Marino","year":"2017"},{"issue":"2","key":"2026033012250439100_ref152","first-page":"428","article-title":"Sublinear time algorithms for earth mover","volume":"48","author":"Do Ba","year":"2011"},{"issue":"2","key":"2026033012250439100_ref153","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s00526-008-0182-5","article-title":"A new class of transport distances between measures","volume":"34","author":"Dolbeault","year":"2009","journal-title":"Calculus of Variations and Partial Differential Equations"},{"issue":"1-2","key":"2026033012250439100_ref154","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s00440-013-0531-y","article-title":"\u201cMartingale optimal transport and robust hedging in continuous time\u201d.","volume":"160","author":"Dolinsky","year":"2014.","journal-title":"Probability Theory and Related Fields"},{"issue":"1","key":"2026033012250439100_ref155","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1214\/aoms\/1177697802","article-title":"\u201cThe speed of mean Glivenko-Cantelli convergence\u201d","volume":"40","author":"Dudley","year":"1969","journal-title":"Annals of Mathematical Statistics"},{"key":"2026033012250439100_ref156","volume-title":"Arxiv:1612.09585","author":"Dupuy","year":"2016"},{"key":"2026033012250439100_ref157","first-page":"10783","author":"Dvurechenskii","year":"2018.","journal-title":"Advances in Neural Information Processing Systems 31"},{"key":"2026033012250439100_ref158","first-page":"1367","author":"Dvurechensky","year":"2018","journal-title":"Proceedings Of The 35Th International Conference On Machine Learning. Ed. By J. Dy And A. Krause. Vol. 80. Proceedings Of Machine Learning Research"},{"issue":"384","key":"2026033012250439100_ref159","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1080\/01621459.1983.10477029","article-title":"\u201cAn algorithm for restricted least squares regression\u201d.","volume":"78","author":"Dykstra","year":"1983","journal-title":"Journal American Statistical Association."},{"issue":"3","key":"2026033012250439100_ref160","first-page":"975","article-title":"\u201cAn iterative procedure for obtaining","volume":"13","author":"Dykstra","year":"1985","journal-title":"I-projections onto the intersection of convex sets\u201d. Annals of Probability"},{"issue":"3","key":"2026033012250439100_ref161","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/BF01581204","article-title":"\u201cOn the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators\u201d.","volume":"55","author":"Eckstein","year":"1992","journal-title":"Mathematical Programming"},{"key":"2026033012250439100_ref162","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/B978-0-12-663450-1.50017-7","author":"Edwards","year":"1975","journal-title":"Studies in topology. Elsevier."},{"issue":"23","key":"2026033012250439100_ref163","doi-asserted-by":"crossref","first-page":"-","DOI":"10.1016\/j.jcp.2012.07.022","article-title":"\u201cBayesian inference with optimal maps\u201d.","volume":"231","author":"El Moselhy","year":"2012","journal-title":"Journal of Computational Physics."},{"issue":"7","key":"2026033012250439100_ref164","doi-asserted-by":"crossref","first-page":"1858","DOI":"10.1109\/TIT.2003.813506","article-title":"\u201cA new metric for probability distributions\u201d.","volume":"49","author":"Endres","year":"2003.","journal-title":"IEEE Transactions on Information theory"},{"issue":"1","key":"2026033012250439100_ref165","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/08-AIHP306","article-title":"\u201cThe heat equation on manifolds as a gradient flow in the Wasserstein space\u201d.","volume":"46","author":"Erbar","year":"2010","journal-title":"Annales de l\u2019Institut Henri Poincar\u00e9, Probabilit\u00e9s et Statistiques"},{"key":"2026033012250439100_ref166","doi-asserted-by":"crossref","first-page":"1355","DOI":"10.3934\/dcds.2014.34.1355","article-title":"\u201cGradient flow structures for discrete porous medium equations\u201d.","author":"Erbar","year":"2014","journal-title":"Discrete and Continuous Dynamical Systems"},{"key":"2026033012250439100_ref167","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-45515-5","author":"Erlander","year":"1980","journal-title":"Optimal Spatial Interaction and the Gravity Model"},{"key":"2026033012250439100_ref168","author":"Erlander","year":"1990","journal-title":"The Gravity Model in Transportation Analysis: Theory and Extensions"},{"issue":"1-2","key":"2026033012250439100_ref169","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1007\/s10107-017-1172-1","article-title":"\u201cData-driven distributionally robust optimizatio e Wasserstein metric:Performance guarantees and tractable reformulations\u201d.","volume":"171","author":"Esfahani","year":"2018","journal-title":"Mathematical Programming."},{"key":"2026033012250439100_ref170","author":"Essid","year":"2017.","journal-title":"arXiv:1704.08200"},{"key":"2026033012250439100_ref171","author":"Evans","year":"1999","journal-title":"Differential Equations Methods for the Monge-Kantorovich Mass Transfer Problem."},{"issue":"4","key":"2026033012250439100_ref172","doi-asserted-by":"crossref","first-page":"1667","DOI":"10.1090\/S0002-9947-01-02930-0","article-title":"\u201cMonge\u2019s transport problem on a Riemannian manifold\u201d.","volume":"354","author":"Feldman","year":"2002.","journal-title":"Transaction AMS."},{"key":"2026033012250439100_ref173","first-page":"291","article-title":"#x201C;Optimal transport for diffeomorphic registration\u201d.","author":"Feydy","year":"2017","journal-title":"In: Proceedings of MIC- CAI\u201917"},{"key":"2026033012250439100_ref174","author":"Feydy","year":"2019.","journal-title":"Proceedings of the 22th International Conference on Artificial Intel ligence and Statistics."},{"issue":"2","key":"2026033012250439100_ref175","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s00205-008-0212-7","article-title":"\u201cThe optimal partial transport problem\u201d.","volume":"195","author":"Figalli","year":"2010.","journal-title":"Archive for Rational Mechanics and Analysis."},{"key":"2026033012250439100_ref176","first-page":"703","author":"Flamary","year":"2016.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"2026033012250439100_ref177","author":"Ford","year":"1962","journal-title":"Flows in Networks. Princeton University Press."},{"issue":"1","key":"2026033012250439100_ref178","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s00220-015-2435-4","article-title":"\u201cRelating the Bures measure to the Cauchy two-matrix model\u201d.","volume":"342","author":"Forrester","year":"2016","journal-title":"Communications in Mathematical Physics."},{"issue":"3-4","key":"2026033012250439100_ref179","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s00440-014-0583-7","article-title":"\u201cOn the rate of convergence in Wasserstein distance of the empirical measure\u201d","volume":"162","author":"Fournier","year":"2015","journal-title":"Probability Theory and Related Fields."},{"key":"2026033012250439100_ref180","first-page":"17","article-title":"\u201cOn the scaling of multidimensional matrices\u201d","volume":"114","author":"Franklin","year":"1989","journal-title":"Linear Algebra and its Applications."},{"issue":"6886","key":"2026033012250439100_ref181","doi-asserted-by":"crossref","first-page":"260","DOI":"10.1038\/417260a","article-title":"\u201cA reconstruction of the initial conditions of the universe by optimal mass transportation\u201d.","volume":"417","author":"Frisch","year":"2002","journal-title":"Nature"},{"issue":"4","key":"2026033012250439100_ref182","doi-asserted-by":"crossref","first-page":"1692","DOI":"10.1137\/100803092","article-title":"\u201cConvergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampere equation in dimensions two and higher\u201d","volume":"49","author":"Froese","year":"2011","journal-title":"SIAM Journal on Numerical Analysis."},{"key":"2026033012250439100_ref183","first-page":"2053","author":"Frogner","year":"2015.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026033012250439100_ref184","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/0898-1221(76)90003-1","article-title":"\u201cA dual algorithm for the solution of nonlinear variational problems via finite element approximation\u201d.","volume":"2","author":"Gabay","year":"1976","journal-title":"Computers & Mathematics with Applications."},{"key":"2026033012250439100_ref185","doi-asserted-by":"crossref","DOI":"10.23943\/princeton\/9780691172767.001.0001","author":"Galichon","year":"2016","journal-title":"Optimal Transport Methods in Economics"},{"issue":"1","key":"2026033012250439100_ref186","doi-asserted-by":"crossref","first-page":"312","DOI":"10.1214\/13-AAP925","article-title":"\u201cA stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options\u201d.","volume":"24","author":"Galichon","year":"2014","journal-title":"Annals of Applied Probability."},{"key":"2026033012250439100_ref187","first-page":"1487307","author":"Galichon","year":"2009","journal-title":"Tech. rep. Preprint SSRN-"},{"key":"2026033012250439100_ref188","first-page":"1","article-title":"\u201cA Lagrangian scheme a la Brenier for the incompressible Euler equations\u201d","volume":"18","author":"Gallouet","year":"2017","journal-title":"Foundations of Computational Mathematics."},{"issue":"2","key":"2026033012250439100_ref189","doi-asserted-by":"crossref","first-page":"1100","DOI":"10.1137\/16M106666X","article-title":"\u201cA JKO splitting scheme for Kantorovich-Fisher-Rao gradient flows\u201d.","volume":"49","author":"Gallouet","year":"2017","journal-title":"SIAM Journal on Mathematical Analysis."},{"issue":"2","key":"2026033012250439100_ref190","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1007\/BF02392620","article-title":"\u201cThe geometry of optimal transportation\u201d","volume":"177","author":"Gangbo","year":"1996","journal-title":"Acta Mathematica."},{"issue":"1","key":"2026033012250439100_ref191","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1002\/(SICI)1097-0312(199801)51:1<23::AID-CPA2>3.0.CO;2-H","article-title":"\u201cOptimal maps for the multidimensional Monge-Kantorovich problem\u201d.","volume":"51","author":"Gangbo","year":"1998","journal-title":"Communications on Pure and Applied Mathematics."},{"key":"2026033012250439100_ref192","first-page":"79137923.","article-title":"\u201cRobust Hypothesis Testing Using Wasserstein Uncertainty Sets\u201d. In:","author":"GAO","year":"2018.","journal-title":"Advances in Neural Information Processing Systems 31."},{"issue":"1","key":"2026033012250439100_ref193","first-page":"185","volume":"147","author":"&gt;Gelbrich","year":"1990","journal-title":"Wasserstein metric between measures on Euclidean and Hilbert spaces\u201d. Mathematische Nachrichten."},{"key":"2026033012250439100_ref194","article-title":"\u201cSample complexity of Sinkhorn divergences\u201d. In:","author":"Genevay","year":"2019","journal-title":"Proceedings of the 22th International Conference on Artificial Intelligence and Statistics"},{"key":"2026033012250439100_ref195","first-page":"3440","article-title":"\u201cStochastic optimization for large-scale optimal transport\u201d.","author":"Genevay","year":"2016","journal-title":"Advances in Neural Information Processing Systems."},{"key":"2026033012250439100_ref196","author":"Genevay","year":"2017","journal-title":"\u201cGAN and VAE from an optimal transport point of view\u201d. (arXiv preprint arXiv:1706.01807)"},{"key":"2026033012250439100_ref197","first-page":"1608","article-title":"\u201cLearning generative models with Sinkhorn divergences\u201d. In:","author":"Genevay","year":"2018","journal-title":"Proceedings of the 21st International Conference on Artificial Intel ligence and Statistics."},{"key":"2026033012250439100_ref198","author":"Gentil","year":"2015","journal-title":"arXiv:1510.08230."},{"issue":"1","key":"2026033012250439100_ref199","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/1031001","article-title":"\u201cThe evolution of the minimum degree ordering algorithm\u201d.","volume":"31","author":"George","year":"1989","journal-title":"SIAM Review."},{"issue":"3","key":"2026033012250439100_ref200","doi-asserted-by":"crossref","first-page":"033301.","DOI":"10.1063\/1.4915289","volume":"56","author":"Georgiou","year":"2015.","journal-title":"Journal of Mathematical Physics."},{"key":"2026033012250439100_ref201","doi-asserted-by":"crossref","first-page":"286","DOI":"10.5201\/ipol.2013.87","author":"Getreuer","year":"2013","journal-title":"Image Processing On Line. 2013:"},{"issue":"1","key":"2026033012250439100_ref202","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s00205-008-0186-5","article-title":"\u201cThe Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation\u201d.","volume":"194","author":"Gianazza","year":"2009.","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"3","key":"2026033012250439100_ref203","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1111\/j.1751-5823.2002.tb00178.x","article-title":"\u201cOn choosing and bounding probability metrics\u201d.","volume":"70","author":"&gt;Gibbs","year":"2002.","journal-title":"International Statistical Review."},{"key":"2026033012250439100_ref204","author":"Glaunes","year":"2004","journal-title":"Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. Vol. 2."},{"key":"2026033012250439100_ref205","first-page":"513","author":"Glorot","year":"2011","journal-title":"Proceedings of the 28th International Conference on Machine Learning."},{"issue":"R2","key":"2026033012250439100_ref206","first-page":"41","article-title":"\u201cSur l\u2019approximation, par \u00e9l\u00e9ments finis d\u2019ordre un, et la r\u00e9solution, par p\u00e9nalisation-dualit\u00e9 d\u2019une classe de probl\u00e8mes de Dirichlet non lin\u00e9aires\u201d.","volume":"9","author":"Glowinski","year":"1975","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis."},{"issue":"4","key":"2026033012250439100_ref207","first-page":"50:1","article-title":"\u201cPower particles: an incompressible fluid solver based on power diagrams\u201d.","volume":"34","author":"Goes","year":"2015","journal-title":"ACM Transaction Graphics."},{"issue":"4","key":"2026033012250439100_ref208","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1109\/34.491619","article-title":"\u201cA graduated assignment algorithm for graph matching\u201d.","volume":"18","author":"&gt;Gold","year":"1996","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence."},{"issue":"8","key":"2026033012250439100_ref209","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1016\/S0031-3203(98)80010-1","article-title":"\u201cNew algorithms for 2D and 3D point matching: pose estimation and correspondence\u201d.","volume":"31","author":"Gold","year":"1998","journal-title":"Pattern Recognition."},{"key":"2026033012250439100_ref210","doi-asserted-by":"crossref","first-page":"345","DOI":"10.5209\/rev_REMA.2003.v16.n1.16889","article-title":"\u201cA survey on continuous elliptical vector distributions\u201d.","volume":"16","author":"Gomez","year":"2003","journal-title":"Rev. Mat. Complut."},{"issue":"16","key":"2026033012250439100_ref211","doi-asserted-by":"crossref","first-page":"166402","DOI":"10.1103\/PhysRevLett.103.166402","article-title":"\u201cDensity-functional theory for strongly interacting electrons\u201d.","volume":"103","author":"Gori-Giorgi","year":"2009","journal-title":"Physical Review Letters."},{"key":"2026033012250439100_ref212","first-page":"261272","article-title":"\u201cFast Optimal Transport Averaging of Neuroimaging Data\u201d. In:","author":"Gramfort","year":"2015","journal-title":"Information Processing in Medical Imaging - 24th International Conference, IPMI 2015."},{"key":"2026033012250439100_ref213","first-page":"1458","article-title":"\u201cThe pyramid match kernel: discriminative classification with sets of image features\u201d. In:","volume":"&gt;Vol. 2.","author":"Grauman","year":"2005","journal-title":"Tenth IEEE International Conference on Computer Vision."},{"key":"2026033012250439100_ref214","article-title":"\u201cUnsupervised Alignment of Embeddings with Wasserstein Procrustes\u201d. In:","author":"Grave","year":"2019","journal-title":"Proceedings of the 22th International Conference on Artificial Intel ligence and Statistics"},{"key":"2026033012250439100_ref215","first-page":"723","article-title":"\u201cA kernel two-sample test\u201d","author":"&gt;Gretton","year":"2012","journal-title":"Journal of Machine Learning Research."},{"key":"2026033012250439100_ref216","first-page":"513","article-title":"\u201cA kernel method for the two-sample-problem\u201d. In:","author":"Gretton","year":"2007","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"1","key":"2026033012250439100_ref217","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1080\/10556789208805505","article-title":"\u201cAchieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation\u201d.","volume":"1","author":"Griewank","year":"1992.","journal-title":"Optimization Methods and Software."},{"key":"2026033012250439100_ref218","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898717761","author":"Griewank","year":"2008","journal-title":"Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM."},{"key":"2026033012250439100_ref219","author":"&gt;Gromov","year":"2001","journal-title":"Metric Structures for Riemannian and Non-Rieman- nian Spaces. Progress in Mathematics. Birkh\u00e4user."},{"key":"2026033012250439100_ref220","article-title":"\u201cComputational methods for martingale optimal transport problems\u201d.","author":"Guo","year":"2017","journal-title":"arXiv:1710.07911."},{"issue":"3","key":"2026033012250439100_ref221","doi-asserted-by":"crossref","first-page":"448","DOI":"10.1016\/j.jcss.2004.06.003","article-title":"\u201cClassical complexity and quantum entanglement\u201d.","volume":"69","author":"Gurvits","year":"2004.","journal-title":"Journal of Computer and System Sciences."},{"key":"2026033012250439100_ref222","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-43374-5","author":"Guti\u00e9rrez","year":"2016.","journal-title":"The Monge-Ampere Equation. Springer."},{"key":"2026033012250439100_ref223","first-page":"172","article-title":"\u201cOptimal patch assignment for statistically constrained texture synthesis\u201d","author":"Gutierrez","year":"2017","journal-title":"In: International Conference on Scale Space and Variational Methods in Computer Vision. Springer."},{"issue":"1","key":"2026033012250439100_ref224","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0377-0427(00)00403-9","article-title":"\u201cSuccessive overrelaxation (SOR) and related methods\u201d.","volume":"123","author":"Hadjidimos","year":"2000","journal-title":"Journal of Computational and Applied Mathematics."},{"issue":"3","key":"2026033012250439100_ref225","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1023\/B:VISI.0000036836.66311.97","article-title":"\u201cOptimal mass transport for registration and warping\u201d.","volume":"60","author":"Haker","year":"2004","journal-title":"International Journal of Computer Vision."},{"issue":"2","key":"2026033012250439100_ref226","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1090\/S0002-9939-1992-1097344-5","article-title":"\u201cKantorovich-Rubinstein norm and its application in the theory of Lipschitz spaces\u201d.","volume":"115","author":"Hanin","year":"1992","journal-title":"Proceedings of the American Mathematical Society."},{"key":"2026033012250439100_ref227","first-page":"2417","article-title":"\u201cLearning populationlevel diffusions with generative RNNs\u201d.","author":"Hashimoto","year":"2016.","journal-title":"International Conference on Machine Learning."},{"issue":"1","key":"2026033012250439100_ref228","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/BF02096204","article-title":"\u201c\u00dcber die gerade Linie als k\u00fcrzeste Verbindung zweier Punkte\u201d.","volume":"46","author":"Hilbert","year":"1895.","journal-title":"Mathematische Annalen."},{"issue":"1-4","key":"2026033012250439100_ref229","first-page":"224","article-title":"\u201cThe distribution of a product from several sources to numerous localities\u201d.","volume":"20","author":"Hitchcock","year":"1941.","journal-title":"Studies in Applied Mathematics."},{"key":"2026033012250439100_ref230","first-page":"1501","author":"Ho","year":"2017","journal-title":"International Conference on Machine Learning."},{"issue":"3","key":"2026033012250439100_ref231","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1214\/009053607000000677","article-title":"\u201cKernel methods in machine learning\u201d.","volume":"36","author":"Hofmann","year":"2008","journal-title":"Annals of Statistics"},{"key":"2026033012250439100_ref232","doi-asserted-by":"crossref","DOI":"10.1002\/9781118548387","author":"Hosmer Jr","year":"2013","journal-title":"Applied Logistic Regression. Vol. 398. John Wiley & Sons"},{"key":"2026033012250439100_ref233","first-page":"4862","author":"Huang","year":"2016","journal-title":"Advances in Neural Information Processing Systems."},{"key":"2026033012250439100_ref234","author":"Idel","year":"2016.","journal-title":"arXiv:1609.06349"},{"key":"2026033012250439100_ref235","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1145\/1993636.1993720","author":"Indyk","year":"2011","journal-title":"Proceedings of the forty-third annual ACM Symposium on Theory of Computing. ACM."},{"key":"2026033012250439100_ref236","author":"Indyk","year":"2003","journal-title":"3rd International Workshop on Statistical and Computational Theories of Vision"},{"issue":"7","key":"2026033012250439100_ref237","doi-asserted-by":"crossref","first-page":"1723","DOI":"10.1109\/TAC.2012.2183171","article-title":"\u201cDistances and Riemannian metrics for multivariate spectral densities\u201d.","volume":"57","author":"Jiang","year":"2012.","journal-title":"IEEE Transactions on Automatic Control."},{"key":"2026033012250439100_ref238","first-page":"189","author":"Johnson","year":"1984","journal-title":"Conference in Modern Analysis and Probability (New Haven, Conn., 1982). Vol. 26. Contemporary Mathematics."},{"issue":"1","key":"2026033012250439100_ref239","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1137\/S0036141096303359","article-title":"\u201cThe variational formulation of the Fokker-Planck equation\u201d.","volume":"29","author":"Jordan","year":"1998","journal-title":"SIAM Journal on Mathematical Analysis."},{"issue":"2","key":"2026033012250439100_ref240","first-page":"227","article-title":"\u201cOn the transfer of masses (in Russian)\u201d.","volume":"37","author":"Kantorovich","year":"1942.","journal-title":"Doklady Akademii Nauk."},{"key":"2026033012250439100_ref241","first-page":"52","article-title":"\u201cOn a space of totally additive functions\u201d.","volume":"13:","author":"Kantorovich","year":"1958","journal-title":"Vestn Leningrad Universitet."},{"key":"2026033012250439100_ref242","author":"Karcher","year":"2014","journal-title":"arXiv:1407.2087"},{"key":"2026033012250439100_ref243","author":"Karlsson","year":"2016","journal-title":"arXiv:1612.02273."},{"key":"2026033012250439100_ref244","first-page":"2259","author":"Kim","year":"2013.","journal-title":"IEEE International Symposium on Information Theory"},{"issue":"04","key":"2026033012250439100_ref245","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1051\/m2an:1999166","article-title":"\u201cApproximation of parabolic equations using the Wasserstein metric\u201d","volume":"33","author":"Kinderlehrer","year":"1999","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"2026033012250439100_ref246","author":"Kitagawa","year":"2016","journal-title":"\u201cA Newton algorithm for semi-discrete optimal transport\u201d"},{"issue":"1","key":"2026033012250439100_ref247","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1137\/060659624","article-title":"\u201cThe Sinkhorn-Knopp algorithm: convergence and applications\u201d","volume":"30","author":"Knight","year":"2008","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"issue":"3","key":"2026033012250439100_ref248","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1093\/imanum\/drs019","article-title":"\u201cA fast algorithm for matrix balancing\u201d","volume":"33","author":"Knight","year":"2013","journal-title":"IMA Journal of Numerical Analysis"},{"issue":"3","key":"2026033012250439100_ref249","doi-asserted-by":"crossref","first-page":"931","DOI":"10.1137\/110825753","article-title":"\u201cA symmetry preserving algorithm for matrix scaling\u201d","volume":"35","author":"Knight","year":"2014","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"issue":"1","key":"2026033012250439100_ref250","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/BF00934745","article-title":"\u201cOn the optimal mapping of distributions\u201d","volume":"43","author":"Knott","year":"1984","journal-title":"Journal of Optimization Theory and Applications"},{"key":"2026033012250439100_ref251","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/0024-3795(94)90359-X","article-title":"\u201cOn a generalization of cyclic monotonicity and distances among random vectors\u201d","volume":"199","author":"Knott","year":"1994","journal-title":"Linear Algebra and Its Applications"},{"issue":"4","key":"2026033012250439100_ref252","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1109\/MSP.2017.2695801","article-title":"\u201cOptimal mass transport: signal processing and machine-learning applications\u201d","volume":"34","author":"Kolouri","year":"2017","journal-title":"IEEE Signal Processing Magazine"},{"key":"2026033012250439100_ref253","first-page":"5258","volume-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition","author":"Kolouri","year":"2016"},{"issue":"11\/12","key":"2026033012250439100_ref254","doi-asserted-by":"crossref","first-page":"1117","DOI":"10.57262\/ade\/1476369298","article-title":"\u201cA new optimal transport distance on the space of finite Radon measures\u201d","volume":"21","author":"Kondratyev","year":"2016","journal-title":"Advances in Differential Equations"},{"key":"2026033012250439100_ref255","first-page":"136","volume-title":"Econometrica: Journal of the Econometric Society","author":"Koopmans","year":"1949"},{"issue":"3","key":"2026033012250439100_ref256","doi-asserted-by":"crossref","first-page":"1501","DOI":"10.1090\/S0002-9947-2014-06032-7","article-title":"\u201cOptimal transportation with capacity constraints\u201d","volume":"367","author":"Korman","year":"2015","journal-title":"Transactions of the American Mathematical Society"},{"key":"2026033012250439100_ref257","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-24488-9","volume-title":"Combinatorial Optimization","author":"Korte","year":"2012"},{"issue":"3","key":"2026033012250439100_ref258","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1016\/0893-6080(94)90081-7","article-title":"\u201cThe invisible hand algorithm: Solving the assignment problem with statistical physics\u201d","volume":"7","author":"Kosowsky","year":"1994","journal-title":"Neural networks"},{"key":"2026033012250439100_ref259","first-page":"E15","article-title":"\u201cTelefoonverkeersrekening\u201d","volume":"52","author":"Kruithof","year":"1937","journal-title":"De Ingenieur"},{"key":"2026033012250439100_ref260","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1002\/nav.3800020109","article-title":"\u201cThe Hungarian method for the assignment problem\u201d","volume":"2","author":"Kuhn","year":"1955","journal-title":"Naval Research Logistics Quarterly"},{"issue":"4","key":"2026033012250439100_ref261","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1561\/2200000019","article-title":"\u201cMetric learning: a survey\u201d","volume":"5","author":"Kulis","year":"2012","journal-title":"Foundations and Trends in Machine Learning"},{"key":"2026033012250439100_ref262","first-page":"957","volume-title":"International Conference on Machine Learning","author":"Kusner","year":"2015"},{"key":"2026033012250439100_ref263","first-page":"9792","volume-title":"Advances in Neural Information Processing Systems 31","author":"Lacombe","year":"2018"},{"issue":"2","key":"2026033012250439100_ref264","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1137\/16M1068827","article-title":"\u201cMultiscale Nonrigid Point Cloud Registration Using Rotation-Invariant Sliced-Wasserstein Distance via Laplace-Beltrami Eigenmap\u201d","volume":"10","author":"Lai","year":"2017","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026033012250439100_ref265","volume-title":"\u201cHarmonic mappings valued in the Wasserstein space\u201d","author":"Lavenant","year":"2017"},{"key":"2026033012250439100_ref266","first-page":"2169","article-title":"\u201cBeyond bags of features: Spatial pyramid matching for recognizing natural scene categories\u201d","volume":"Vol. 2","author":"Lazebnik","year":"2006","journal-title":"IEEE Computer Society Conference on Computer Vision and Pattern Recognition"},{"key":"2026033012250439100_ref267","doi-asserted-by":"crossref","first-page":"901","DOI":"10.1007\/s00440-016-0727-z","article-title":"\u201cExistence and consistency of Wasserstein barycenters\u201d","volume":"168","author":"Le Gouic","year":"2016","journal-title":"Probability Theory and Related Fields"},{"issue":"6755","key":"2026033012250439100_ref268","doi-asserted-by":"crossref","first-page":"788","DOI":"10.1038\/44565","article-title":"\u201cLearning the parts of objects by non-negative matrix factorization\u201d","volume":"401","author":"Lee","year":"1999","journal-title":"Nature"},{"key":"2026033012250439100_ref269","first-page":"2692","volume-title":"Advances in Neural Information Processing Systems 31","author":"Lee","year":"2018"},{"issue":"4","key":"2026033012250439100_ref270","doi-asserted-by":"crossref","first-page":"910","DOI":"10.1007\/s00041-015-9439-5","article-title":"\u201cHolder-Lipschitz norms and their duals on spaces with semigroups, with applications to earth mover\u2019s distance\u201d","volume":"22","author":"Leeb","year":"2016","journal-title":"Journal of Fourier Analysis and Applications"},{"issue":"4","key":"2026033012250439100_ref271","doi-asserted-by":"crossref","first-page":"2833","DOI":"10.1137\/140975528","article-title":"\u201cImaging with Kantorovich-Rubinstein discrepancy\u201d","volume":"7","author":"Lellmann","year":"2014","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026033012250439100_ref272","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139026079","volume":"Vol. 189","author":"Lemmens","year":"2012","journal-title":"Nonlinear Perron-Frobenius Theory"},{"issue":"4","key":"2026033012250439100_ref273","doi-asserted-by":"crossref","first-page":"1879","DOI":"10.1016\/j.jfa.2011.11.026","article-title":"\u201cFrom the Schrodinger problem to the Monge- Kantorovich problem\u201d","volume":"262","author":"L\u00e9onard","year":"2012","journal-title":"Journal of Functional Analysis"},{"issue":"4","key":"2026033012250439100_ref274","doi-asserted-by":"crossref","first-page":"1533","DOI":"10.3934\/dcds.2014.34.1533","article-title":"\u201cA survey of the Schrodinger problem and some of its connections with optimal transport\u201d","volume":"34","author":"L\u00e9onard","year":"2014","journal-title":"Discrete Continuous Dynamical Systems Series A"},{"issue":"6","key":"2026033012250439100_ref275","doi-asserted-by":"crossref","first-page":"1693","DOI":"10.1051\/m2an\/2015055","article-title":"\u201cA numerical algorithm for L2 semi-discrete optimal transport in 3D\u201d","volume":"49","author":"L\u00e9vy","year":"2015","journal-title":"ESAIM: Mathematical Model ling and Numerical Analysis"},{"key":"2026033012250439100_ref276","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.cag.2018.01.009","article-title":"\u201cNotions of optimal transport theory and how to implement them on a computer\u201d","volume":"72","author":"L\u00e9vy","year":"2018","journal-title":"Computers & Graphics"},{"key":"2026033012250439100_ref278","first-page":"1689","volume-title":"Proceedings of the IEEE International Conference on Computer Vision","author":"Li","year":"2013"},{"issue":"1","key":"2026033012250439100_ref279","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1007\/s10915-017-0529-1","article-title":"\u201cA parallel method for Earth Mover\u2019s distance\u201d","volume":"75","author":"Li","year":"2018","journal-title":"Journal of Scientific Computing"},{"key":"2026033012250439100_ref280","volume-title":"\u201cImage segmentation via L1 Monge- Kantorovich problem\u201d","author":"Li","year":"2018"},{"issue":"4","key":"2026033012250439100_ref281","doi-asserted-by":"crossref","first-page":"2869","DOI":"10.1137\/15M1041420","article-title":"\u201cOptimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves\u201d","volume":"48","author":"Liero","year":"2016","journal-title":"SIAM Journal on Mathematical Analysis"},{"issue":"3","key":"2026033012250439100_ref282","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1007\/s00222-017-0759-8","article-title":"\u201cOptimal entropy-transport problems and a new Hellinger-Kantorovich distance between positive measures\u201d","volume":"211","author":"Liero","year":"2018","journal-title":"Inventiones Mathematicae"},{"key":"2026033012250439100_ref283","first-page":"246","article-title":"\u201cDiffusion distance for histogram comparison\u201d","volume":"Vol. 1","author":"Ling","year":"2006","journal-title":"IEEE Computer Society Conference on Computer Vision and Pattern Recognition"},{"issue":"5","key":"2026033012250439100_ref284","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1109\/TPAMI.2007.1058","article-title":"\u201cAn efficient earth mover\u2019s distance algorithm for robust histogram comparison\u201d","volume":"29","author":"Ling","year":"2007","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"2026033012250439100_ref285","doi-asserted-by":"crossref","first-page":"644","DOI":"10.1145\/276698.276880","volume-title":"Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing","author":"Linial","year":"1998"},{"key":"2026033012250439100_ref286","doi-asserted-by":"crossref","first-page":"964","DOI":"10.1137\/0716071","article-title":"\u201cSplitting algorithms for the sum of two nonlinear operators\u201d","volume":"16","author":"Lions","year":"1979","journal-title":"SIAM Journal on Numerical Analysis"},{"issue":"3","key":"2026033012250439100_ref287","doi-asserted-by":"crossref","first-page":"1049","DOI":"10.1214\/aoms\/1177700079","article-title":"\u201cA nonparametric estimate of a multivariate density function\u201d","volume":"36","author":"Loftsgaarden","year":"1965","journal-title":"Annals of Mathematical Statistics"},{"issue":"2","key":"2026033012250439100_ref288","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1016\/j.ejor.2005.09.032","article-title":"\u201cA survey for the quadratic assignment problem\u201d","volume":"176","author":"Loiola","year":"2007","journal-title":"European Journal Operational Research"},{"issue":"1","key":"2026033012250439100_ref289","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1109\/TPAMI.2015.2424894","article-title":"\u201cGraph matching: relax at your own risk\u201d","volume":"38","author":"Lyzinski","year":"2016","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intel ligence"},{"issue":"8","key":"2026033012250439100_ref290","doi-asserted-by":"crossref","first-page":"2250","DOI":"10.1016\/j.jfa.2011.06.009","article-title":"\u201cGradient flows of the entropy for finite Markov chains\u201d","volume":"261","author":"Maas","year":"2011","journal-title":"Journal of Functional Analysis"},{"issue":"6","key":"2026033012250439100_ref291","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1051\/m2an\/2015043","article-title":"\u201cA generalized model for optimal transport of images including dissipation and density modulation\u201d","volume":"49","author":"Maas","year":"2015","journal-title":"ESAIM: Mathematical Model ling and Numerical Analysis"},{"key":"2026033012250439100_ref292","volume-title":"\u201cGeneralized optimal transport with singular sources\u201d","author":"Maas","year":"2016"},{"key":"2026033012250439100_ref293","first-page":"202","volume-title":"Asian Conference on Computer Vision","author":"Makihara","year":"2010"},{"key":"2026033012250439100_ref294","volume-title":"\u201cWasserstein Riemannian geometry of positive-definite matrices\u201d","author":"Malago","year":"2018"},{"key":"2026033012250439100_ref295","first-page":"5660","volume-title":"Advances in Neural Information Processing Systems 30","author":"Mallasto","year":"2017"},{"key":"2026033012250439100_ref296","volume-title":"A Wavelet Tour of Signal Processing: the Sparse Way","author":"Mallat","year":"2008"},{"issue":"4","key":"2026033012250439100_ref297","doi-asserted-by":"crossref","first-page":"1694","DOI":"10.1109\/TIP.2014.2305873","article-title":"\u201cOptimal transport for secure spread-spectrum watermarking of still images\u201d","volume":"23","author":"Mathon","year":"2014","journal-title":"IEEE Transactions on Image Processing"},{"issue":"3","key":"2026033012250439100_ref298","doi-asserted-by":"crossref","first-page":"697","DOI":"10.1051\/m2an\/2013126","article-title":"\u201cConvergence of a variational Lagrangian scheme for a nonlinear drift diffusion equation\u201d","volume":"48","author":"Matthes","year":"2014","journal-title":"ESAIM: Mathematical Model ling and Numerical Analysis"},{"issue":"1","key":"2026033012250439100_ref299","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s10208-015-9284-6","article-title":"\u201cA convergent Lagrangian discretization for a nonlinear fourth-order equation\u201d","volume":"17","author":"Matthes","year":"2017.","journal-title":"Foundations of Computational Mathematics"},{"key":"2026033012250439100_ref300","volume-title":"Topological Optimization and Optimal Transport: In the Applied Sciences","author":"Maury","year":"2017"},{"issue":"10","key":"2026033012250439100_ref301","doi-asserted-by":"crossref","first-page":"1787","DOI":"10.1142\/S0218202510004799","article-title":"#x201C;A macroscopic crowd motion model of gradient flow type\u201d","volume":"20","author":"Maury","year":"2010","journal-title":"Mathematical Models and Methods in Applied Sciences"},{"issue":"1","key":"2026033012250439100_ref302","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1006\/aima.1997.1634","article-title":"\u201cA convexity principle for interacting gases\u201d","volume":"128","author":"&gt;McCann","year":"1997","journal-title":"Advances in Mathematics"},{"key":"2026033012250439100_ref303","first-page":"81","volume-title":"Symposium on Point Based Graphics","author":"Memoli","year":"2007"},{"issue":"4","key":"2026033012250439100_ref304","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s10208-011-9093-5","article-title":"\u201cGromov-Wasserstein distances and the metric approach to object matching\u201d","volume":"11","author":"Memoli","year":"2011","journal-title":"Foundations of Computational Mathematics"},{"issue":"3","key":"2026033012250439100_ref305","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/s10208-004-0145-y","article-title":"\u201cA theoretical and computational framework for isometry invariant recognition of point cloud data\u201d","volume":"5","author":"M\u00e9moli","year":"2005","journal-title":"Foundations of Computational Mathematics"},{"issue":"5","key":"2026033012250439100_ref306","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1111\/j.1467-8659.2011.02032.x","article-title":"\u201cA multiscale approach to optimal transport\u201d","volume":"30","author":"M\u00e9rigot","year":"2011","journal-title":"Computer Graphics Forum"},{"issue":"11","key":"2026033012250439100_ref307","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1088\/0266-5611\/32\/11\/115008","article-title":"\u201cAn optimal transport approach for seismic tomography: Application to 3D full waveform inversion\u201d","volume":"32","author":"M\u00e9tivier","year":"2016","journal-title":"Inverse Problems"},{"key":"2026033012250439100_ref308","first-page":"224","volume-title":"SIGGRAPH Asia 2018 Technical Papers","author":"Meyron","year":"2018"},{"issue":"1-2","key":"2026033012250439100_ref309","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00526-012-0538-8","article-title":"\u201cGeodesic convexity of the relative entropy in reversible Markov chains\u201d","volume":"48","author":"Mielke","year":"2013","journal-title":"Calculus of Variations and Partial Differential Equations"},{"key":"2026033012250439100_ref310","volume-title":"\u201cEfficient estimation of word representations in vector space\u201d","author":"Mikolov","year":"2013"},{"issue":"5","key":"2026033012250439100_ref311","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1051\/m2an\/2015016","article-title":"\u201cDiscretization of the 3D Monge-Ampere operator, between wide stencils and power diagrams\u201d","volume":"49","author":"Mirebeau","year":"2015","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"2026033012250439100_ref312","first-page":"666","volume-title":"Histoire de l\u2019Acad\u00e9mie Royale des Sciences","author":"Monge"},{"key":"2026033012250439100_ref313","first-page":"3718","volume-title":"Advances in Neural Information Processing Systems 29","author":"Montavon","year":"2016"},{"key":"2026033012250439100_ref0314","first-page":"2420","volume-title":"Advances in Neural Information Processing Systems","author":"Moon","year":"2014"},{"issue":"4","key":"2026033012250439100_ref315","doi-asserted-by":"crossref","first-page":"1068","DOI":"10.1137\/080721522","article-title":"\u201cOn the application of the Monge-Kantorovich problem to image registration\u201d","volume":"2","author":"Museyko","year":"2009","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026033012250439100_ref316","first-page":"10258","volume-title":"Advances in Neural Information Processing Systems 31","author":"Muzellec","year":"2018"},{"key":"2026033012250439100_ref317","first-page":"2387","author":"Muzellec","year":"2017"},{"issue":"12","key":"2026033012250439100_ref318","doi-asserted-by":"crossref","first-page":"2262","DOI":"10.1109\/TPAMI.2010.46","article-title":"\u201cPoint set registration: coherent pointdrift\u201d","volume":"32","author":"Myronenko","year":"2010","journal-title":"IEEE Transactionson Pattern Analysis and Machine Intel ligence"},{"issue":"3","key":"2026033012250439100_ref319","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1137\/05064206X","volume":"37","author":"Naor","year":"2007","journal-title":"SIAM Journal on Computing"},{"issue":"3","key":"2026033012250439100_ref320","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1137\/080743627","article-title":"\u201cIntroduction to automatic differentiation and Matlab object-oriented programming\u201d","volume":"52","author":"Neidinger","year":"2010.","journal-title":"SIAM Review"},{"key":"2026033012250439100_ref321","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1016\/S0024-3795(99)00212-8","article-title":"\u201cOn complexity of matrix scaling\u201d","volume":"302","author":"Nemirovski","year":"1999","journal-title":"Linear Algebra and its Applications"},{"key":"2026033012250439100_ref322","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970791","volume":"Vol. 13.","author":"Nesterov","year":"1994","journal-title":"Interior-point polynomial algorithms in convex programming"},{"issue":"1","key":"2026033012250439100_ref323","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1007\/s11263-009-0234-0","article-title":"\u201cLocal histogram based segmentation using the Wasserstein distance\u201d","volume":"84","author":"Ni","year":"2009","journal-title":"International Journal of Computer Vision"},{"key":"2026033012250439100_ref324","doi-asserted-by":"crossref","first-page":"2642","DOI":"10.1109\/CDC.2014.7039793","volume-title":"53rd IEEE Conference on Decision and Control","author":"Ning","year":"2014"},{"issue":"2","key":"2026033012250439100_ref325","doi-asserted-by":"crossref","first-page":"373","DOI":"10.1109\/TAC.2014.2350171","article-title":"\u201cOn matrix-valued Monge-Kantorovich optimal mass transport\u201d","volume":"60","author":"Ning","year":"2015","journal-title":"IEEE Transactions on Automatic Control"},{"key":"2026033012250439100_ref326","doi-asserted-by":"crossref","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"Nocedal","year":"1999"},{"key":"2026033012250439100_ref327","volume-title":"\u201cAn efficient linear programming method for optimal transportation\u201d","author":"Oberman","year":"2015"},{"issue":"3","key":"2026033012250439100_ref329","first-page":"271","volume":"54","author":"Oliker","year":"1989"},{"issue":"3","key":"2026033012250439100_ref330","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1023\/A:1011139631724","article-title":"\u201cModeling the shape of the scene: a holistic representation of the spatial envelope\u201d","volume":"42","author":"Oliva","year":"2001","journal-title":"International Journal of Computer Vision"},{"key":"2026033012250439100_ref331","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jcp.2014.01.048","article-title":"\u201cMinimization for conditional simulation: Relationship to optimal transport\u201d","volume":"265","author":"Oliver","year":"2014","journal-title":"Journal of Computational Physics"},{"issue":"2","key":"2026033012250439100_ref332","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/BF02614365","article-title":"\u201cA polynomial time primal network simplex algorithm for minimum cost flows\u201d","volume":"78","author":"Orlin","year":"1997","journal-title":"Mathematical Programming"},{"issue":"3","key":"2026033012250439100_ref333","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1007\/BF02517812","article-title":"\u201cA new class of metric divergences on probability spaces and its applicability in statistics\u201d","volume":"55","author":"\u00d6sterreicher","year":"2003","journal-title":"Annals of the Institute of Statistical Mathematics"},{"issue":"1-2","key":"2026033012250439100_ref334","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1081\/PDE-100002243","article-title":"\u201cThe geometry of dissipative evolution equations: the porous medium equation\u201d","volume":"26","author":"Otto","year":"2001","journal-title":"Communications in Partial Differential Equations"},{"key":"2026033012250439100_ref335","volume-title":"Empirical Likelihood","author":"Owen","year":"2001"},{"issue":"10","key":"2026033012250439100_ref336","doi-asserted-by":"crossref","first-page":"1345","DOI":"10.1109\/TKDE.2009.191","article-title":"\u201cA survey on transfer learning\u201d","volume":"22","author":"Pan","year":"2010","journal-title":"IEEE Transactions on knowledge and data engineering"},{"issue":"2","key":"2026033012250439100_ref337","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1214\/15-AOS1387","article-title":"\u201cAmplitude and phase variation of point processes\u201d","volume":"44","author":"Panaretos","year":"2016","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026033012250439100_ref338","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1137\/130920058","article-title":"\u201cOptimal transport with proximal splitting\u201d","volume":"7","author":"Papadakis","year":"2014","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"3-4","key":"2026033012250439100_ref339","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s00526-011-0421-z","article-title":"\u201cOn the local structure of optimal measures in the multimarginal optimal transportation problem\u201d","volume":"43","author":"Pass","year":"2012","journal-title":"Calculus of Variations and Partial Differential Equations"},{"issue":"6","key":"2026033012250439100_ref340","doi-asserted-by":"crossref","first-page":"1771","DOI":"10.1051\/m2an\/2015020","article-title":"\u201cMulti-marginal optimal transport: theory and applications\u201d","volume":"49","author":"Pass","year":"2015","journal-title":"ESAIM: Mathematical Model ling and Numerical Analysis"},{"key":"2026033012250439100_ref341","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1007\/978-3-642-40020-9_43","volume-title":"Geometric Science of Information","author":"Pele","year":"2013"},{"key":"2026033012250439100_ref342","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1007\/978-3-540-88690-7_37","volume-title":"Computer Vision-ECCV 2008","author":"Pele","year":"2008"},{"key":"2026033012250439100_ref343","first-page":"460","volume-title":"IEEE 12th International Conference on Computer Vision","author":"Pele","year":"2009"},{"issue":"1","key":"2026033012250439100_ref344","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1007\/s00205-013-0704-y","article-title":"\u201cThe Hele-Shaw asymptotics for mechanical models of tumor growth\u201d","volume":"212","author":"&gt;Perthame","year":"2014","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"4","key":"2026033012250439100_ref345","doi-asserted-by":"crossref","first-page":"2323","DOI":"10.1137\/15M1010087","article-title":"\u201cEntropic approximation of Wasserstein gradient flows\u201d","volume":"8","author":"Peyr\u00e9","year":"2015","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"2026033012250439100_ref346","volume-title":"to appear in European Journal of Applied Mathematics","author":"Peyr\u00e9","year":"2017"},{"key":"2026033012250439100_ref347","first-page":"2664","volume-title":"International Conference on Machine Learning","author":"Peyr\u00e9","year":"2016"},{"key":"2026033012250439100_ref348","first-page":"2541","author":"Peyr\u00e9","year":"2012","journal-title":"19th IEEE International Conference on Image Processing"},{"key":"2026033012250439100_ref349","author":"Peyre","year":"2011"},{"issue":"1","key":"2026033012250439100_ref350","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1007\/s00205-013-0669-x","article-title":"\u201cGeneralized Wasserstein distance and its application to transport equations with source\u201d","volume":"211","author":"Piccoli","year":"2014","journal-title":"Archive for Rational Mechanics and Analysis"},{"issue":"1","key":"2026033012250439100_ref351","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.cviu.2006.11.011","article-title":"\u201cAutomated colour grading using colour distribution transfer\u201d","volume":"107","author":"Piti\u00e9","year":"2007","journal-title":"Computer Vision and Image Understanding"},{"key":"2026033012250439100_ref352","first-page":"256","volume-title":"Proceedings of SSVM\u201915","author":"Rabin","year":"2015"},{"key":"2026033012250439100_ref353","first-page":"435","volume-title":"International Conference on Scale Space and Variational Methods in Computer Vision","author":"Rabin","year":"2011"},{"key":"2026033012250439100_ref354","volume-title":"Mass Transportation Problems: Volume I: Theory","author":"Rachev","year":"1998"},{"key":"2026033012250439100_ref355","volume-title":"Mass Transportation Problems: Volume II: Applications","author":"Rachev","year":"1998"},{"key":"2026033012250439100_ref356","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-10861-0","author":"Rall","year":"1981","journal-title":"Automatic Differentiation: Techniques and Applications"},{"issue":"2","key":"2026033012250439100_ref357","doi-asserted-by":"crossref","first-page":"47","DOI":"10.3390\/e19020047","article-title":"\u201cOn Wasserstein two- sample testing and related families of nonparametric tests\u201d","volume":"19","author":"Ramdas","year":"2017","journal-title":"Entropy"},{"issue":"6","key":"2026033012250439100_ref358","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1162\/089976699300016313","article-title":"\u201cConvergence properties of the softassign quadratic assignment algorithm\u201d","volume":"11","author":"Rangarajan","year":"1999","journal-title":"Neural Computation"},{"issue":"4","key":"2026033012250439100_ref359","doi-asserted-by":"crossref","first-page":"A2013","DOI":"10.1137\/130907367","article-title":"#x201C;A nonparametric ensemble transform method for Bayesian inference\u201d","volume":"35","author":"Reich","year":"2013","journal-title":"SIAM Journal on Scientific Computing"},{"issue":"5","key":"2026033012250439100_ref360","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1137\/0314056","article-title":"\u201cMonotone operators and the proximal point algorithm\u201d","volume":"14","author":"Rockafellar","year":"1976","journal-title":"SIAM Journal on Control and Optimization"},{"key":"2026033012250439100_ref361","first-page":"630","article-title":"\u201cFast dictionary learning with a smoothed Wasserstein loss\u201d","volume":"Vol. 51","author":"Rolet","year":"2016","journal-title":"Proceedings of the 19th International Conference on Artificial Intel ligence and Statistics"},{"issue":"2","key":"2026033012250439100_ref362","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1023\/A:1026543900054","article-title":"\u201cThe earth mover\u2019s distance as a metric for image retrieval\u201d","volume":"40","author":"Rubner","year":"2000","journal-title":"International Journal of Computer Vision"},{"issue":"4","key":"2026033012250439100_ref363","doi-asserted-by":"crossref","first-page":"1160","DOI":"10.1214\/aos\/1176324703","article-title":"\u201cConvergence of the iterative proportional fitting Procedure\u201d","volume":"23","author":"Ruschendorf","year":"1995","journal-title":"Annals of Statistics"},{"issue":"1","key":"2026033012250439100_ref364","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/0047-259X(90)90070-X","article-title":"\u201cA characterization of random variables with minimum L2-distance\u201d","volume":"32","author":"Ruschendorf","year":"1990","journal-title":"Journal of Multivariate Analysis"},{"issue":"3","key":"2026033012250439100_ref365","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1137\/S0040585X97976301","article-title":"\u201cClosedness of sum spaces and the generalized Schrodinger problem\u201d","volume":"42","author":"Ruschendorf","year":"1998","journal-title":"Theory of Probability and its Applications"},{"issue":"2","key":"2026033012250439100_ref366","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1006\/jmva.2001.2005","article-title":"\u201cOn the n-coupling problem\u201d","volume":"81","author":"Ruschendorf","year":"2002","journal-title":"Journal of Multivariate Analysis"},{"issue":"5","key":"2026033012250439100_ref367","doi-asserted-by":"crossref","first-page":"A3675","DOI":"10.1137\/17M1163396","article-title":"\u201cVector and matrix optimal mass transport: theory, algorithm, and applications\u201d","volume":"40","author":"Ryu","year":"2017","journal-title":"SIAM Journal on Scientific Compututing"},{"issue":"3","key":"2026033012250439100_ref368","doi-asserted-by":"crossref","first-page":"1596","DOI":"10.1007\/s10915-017-0600-y","article-title":"\u201cUnbalanced and partial L1 Monge-Kantorovich problem: a scalable parallel firstorder method\u201d","volume":"75","author":"Ryu","year":"2017","journal-title":"Journal of Scientific Computing"},{"key":"2026033012250439100_ref369","author":"Salimans","year":"2018","journal-title":"International Conference on Learning Representations"},{"issue":"1","key":"2026033012250439100_ref370","first-page":"57","volume":"4","author":"Samelson","year":"1957","journal-title":"\u201cOn the Perron-Frobenius theorem.\u201d"},{"issue":"8","key":"2026033012250439100_ref371","doi-asserted-by":"crossref","first-page":"1590","DOI":"10.1109\/TPAMI.2011.18","article-title":"\u201cNonnegative matrix factorization with earth mover\u2019s distance metric for image analysis\u201d","volume":"33","author":"Sandler","year":"2011","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intel ligence"},{"key":"2026033012250439100_ref372","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-20828-2","author":"Santambrogio","year":"2015","journal-title":"Optimal transport for applied mathematicians"},{"issue":"1","key":"2026033012250439100_ref373","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1007\/s13373-017-0101-1","article-title":"\u201c{Euclidean, metric, and Wasserstein} gradient flows: an overview\u201d","volume":"7","author":"Santambrogio","year":"2017","journal-title":"Bulletin of Mathematical Sciences"},{"key":"2026033012250439100_ref374","author":"Santambrogio","year":"2018","journal-title":"GMT preprint 3728"},{"issue":"1","key":"2026033012250439100_ref375","doi-asserted-by":"crossref","first-page":"269","DOI":"10.4310\/CMS.2015.v13.n1.a13","article-title":"\u201cOptimal transport for particle image velocimetry\u201d","volume":"13","author":"Saumier","year":"2015","journal-title":"Communications in Mathematical Sciences"},{"key":"2026033012250439100_ref376","first-page":"191","author":"Schiebinger","year":"2017","journal-title":"\u201cReconstruction of developmental landscapes by optimal-transport analysis of single-cell gene expression sheds light on cellular reprogramming.\u201d"},{"issue":"2","key":"2026033012250439100_ref377","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1007\/s10851-016-0653-9","article-title":"\u201cA sparse multiscale algorithm for dense optimal transport\u201d","volume":"56","author":"Schmitzer","year":"2016","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2026033012250439100_ref378","author":"Schmitzer","year":"2016","journal-title":"\u201cStabilized sparse scaling algorithms for entropy regularized transport problems\u201d"},{"issue":"1","key":"2026033012250439100_ref379","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s10851-012-0375-6","article-title":"\u201cModelling convex shape priors and matching based on the Gromov-Wasserstein distance\u201d","volume":"46","author":"Schmitzer","year":"2013","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2026033012250439100_ref380","first-page":"123","author":"Schmitzer","year":"2013","journal-title":"International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition"},{"key":"2026033012250439100_ref381","author":"Schmitzer","year":"2017","journal-title":"\u201cA framework for Wasserstein-1-type metrics\u201d"},{"key":"2026033012250439100_ref382","doi-asserted-by":"crossref","first-page":"522","DOI":"10.1090\/S0002-9947-1938-1501980-0","article-title":"\u201cMetric spaces and positive definite functions\u201d","volume":"38","author":"Schoenberg","year":"1938","journal-title":"Transactions of the American Mathematical Society"},{"key":"2026033012250439100_ref383","volume-title":"Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond","author":"Sch\u00f6lkopf","year":"2002"},{"key":"2026033012250439100_ref384","first-page":"144","volume":"144","author":"Schr\u00f6dinger","year":"1931","journal-title":"\u201c\u00dcber die Umkehrung der Naturgesetze\u201d"},{"key":"2026033012250439100_ref385","first-page":"3294","author":"Seguy","year":"2015","journal-title":"Advances in Neural Information Processing Systems 28"},{"key":"2026033012250439100_ref386","author":"Seguy","year":"2018","journal-title":"Proceedings of ICLR 2018"},{"key":"2026033012250439100_ref387","first-page":"1576","author":"Shafieezadeh Abadeh","year":"2015","journal-title":"Advances in Neural Information Processing Systems 28"},{"key":"2026033012250439100_ref388","first-page":"8483","author":"Shafieezadeh Abadeh","year":"2018","journal-title":"Advances in Neural Information Processing Systems 31"},{"key":"2026033012250439100_ref389","first-page":"1","volume-title":"IEEE Conference on Computer Vision and Pattern Recognition","author":"Shirdhonkar","year":"2008"},{"key":"2026033012250439100_ref390","volume":"Vol. 26","author":"Silverman","year":"1986","journal-title":"Density Estimation for Statistics and Data Analysis"},{"key":"2026033012250439100_ref391","doi-asserted-by":"crossref","first-page":"876","DOI":"10.1214\/aoms\/1177703591","article-title":"\u201cA relationship between arbitrary positive matrices and doubly stochastic matrices\u201d","volume":"35","author":"Sinkhorn","year":"1964","journal-title":"Annals of Mathematical Statistis- tics"},{"issue":"2","key":"2026033012250439100_ref392","doi-asserted-by":"crossref","first-page":"211","DOI":"10.15803\/ijnc.1.2_211","article-title":"\u201cGPU-based softAssign for maximizing image utilization in photomosaics\u201d","volume":"1","author":"Slomp","year":"2011","journal-title":"International Journal of Networking and Computing"},{"issue":"4","key":"2026033012250439100_ref393","doi-asserted-by":"crossref","first-page":"66:1","DOI":"10.1145\/2766963","article-title":"\u201cConvolutional Wasserstein distances: efficient optimal transportation on geometric domains\u201d","volume":"34","author":"Solomon","year":"2015","journal-title":"ACM Transactions on Graphics"},{"issue":"No. 5.","key":"2026033012250439100_ref394","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1111\/cgf.12186","article-title":"\u201cDirichlet energy for analysis and synthesis of soft maps\u201d","volume":"Vol. 32.","author":"Solomon","year":"2013","journal-title":"Computer Graphics Forum"},{"issue":"4","key":"2026033012250439100_ref395","doi-asserted-by":"crossref","first-page":"72:1","DOI":"10.1145\/2897824.2925903","article-title":"\u201cEntropic metric alignment for correspondence problems\u201d","volume":"35","author":"Solomon","year":"2016","journal-title":"ACM Transactions on Graphics"},{"issue":"4","key":"2026033012250439100_ref396","article-title":"\u201cEarth mover\u2019s distances on discrete surfaces\u201d","volume":"33","author":"Solomon","year":"2014","journal-title":"Transaction on Graphics"},{"key":"2026033012250439100_ref397","volume-title":"\u201cContinuous-flow graph transportation distances\u201d","author":"Solomon","year":"2016"},{"key":"2026033012250439100_ref398","first-page":"306","volume-title":"Proceedings of the 31st International Conference on Machine Learning","author":"Solomon","year":"2014"},{"issue":"1","key":"2026033012250439100_ref399","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1111\/rssb.12236","article-title":"\u201cInference for empirical Wasserstein distances on finite spaces\u201d","volume":"80","author":"Sommerfeld","year":"2018","journal-title":"Journal of the Royal Statistical Society: Series B (Statistical Methodology)"},{"key":"2026033012250439100_ref400","volume-title":"\u201cOn integral probability metrics,ip-divergences and binary classification\u201d","author":"Sriperumbudur","year":"2009"},{"key":"2026033012250439100_ref401","doi-asserted-by":"crossref","first-page":"1550","DOI":"10.1214\/12-EJS722","article-title":"\u201cOn the empirical estimation of integral probability metrics\u201d","volume":"6","author":"Sriperumbudur","year":"2012","journal-title":"Electronic Journal of Statistics"},{"key":"2026033012250439100_ref402","first-page":"912","volume-title":"Artificial Intelligence and Statistics","author":"Srivastava","year":"2015"},{"key":"2026033012250439100_ref403","first-page":"2647","volume-title":"Advances in Neural Information Processing Systems 30","author":"Staib","year":"2017"},{"key":"2026033012250439100_ref404","first-page":"2647","volume-title":"Advances in Neural Information Processing Systems 30","author":"Staib","year":"2017"},{"key":"2026033012250439100_ref405","volume-title":"\u201cA polynomial bound on the diameter of the transportation polytope\u201d","author":"Stougie","year":"2002"},{"key":"2026033012250439100_ref406","volume-title":"\u201cThe space of spaces: curvature bounds and gradient flows on the space of metric measure spaces\u201d","author":"Sturm","year":"2012"},{"issue":"11","key":"2026033012250439100_ref407","doi-asserted-by":"crossref","first-page":"2246","DOI":"10.1109\/TPAMI.2015.2408346","article-title":"\u201cOptimal mass transport for shape matching and comparison\u201d","volume":"37","author":"Su","year":"2015","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"2026033012250439100_ref408","volume-title":"Geometric Problems in the Theory of Infinitedimensional Probability Distributions","author":"Sudakov","year":"1979"},{"key":"2026033012250439100_ref409","volume-title":"\u201cTensor balancing on statistical manifold\u201d","author":"Sugiyama","year":"2017"},{"issue":"3","key":"2026033012250439100_ref410","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/j.apnum.2010.10.006","article-title":"\u201cAn efficient approach for the numerical solution of the Monge-Ampere equation\u201d","volume":"61","author":"Sulman","year":"2011","journal-title":"Applied Numerical Mathematics"},{"issue":"3","key":"2026033012250439100_ref411","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.1137\/120897535","article-title":"\u201cConvex variational image restoration with histogram priors\u201d","volume":"6","author":"Swoboda","year":"2013","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"16.10","key":"2026033012250439100_ref412","article-title":"\u201cTesting for equal distributions in high dimension\u201d","volume":"5","author":"Szekely","year":"2004","journal-title":"InterStat"},{"issue":"4","key":"2026033012250439100_ref413","first-page":"1005","article-title":"\u201cWasserstein geometry of Gaussian measures\u201d","volume":"48","author":"Takatsu","year":"2011","journal-title":"Osaka Journal of Mathematics"},{"issue":"5","key":"2026033012250439100_ref414","doi-asserted-by":"crossref","first-page":"3201","DOI":"10.1214\/12-AOP797","article-title":"\u201cOptimal transportation under controlled stochastic dynamics\u201d","volume":"41","author":"Tan","year":"2013","journal-title":"Annals of Probability"},{"issue":"2","key":"2026033012250439100_ref415","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1007\/BF02614369","article-title":"\u201cDynamic trees as search trees via Euler tours, applied to the network simplex algorithm\u201d","volume":"78","author":"Tarjan","year":"1997","journal-title":"Mathematical Programming"},{"issue":"4","key":"2026033012250439100_ref416","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1137\/16M1067494","article-title":"#x201C;Wasserstein loss for image synthesis and restoration\u201d","volume":"9","author":"Tartavel","year":"2016","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"2","key":"2026033012250439100_ref417","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/s10851-017-0726-4","article-title":"\u201cA transportation Lp distance for signal analysis\u201d","volume":"59","author":"Thorpe","year":"2017","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2026033012250439100_ref418","first-page":"23","volume-title":"TransPress of the National Commissariat of Transportation","author":"Tolstoi","year":"1930"},{"key":"2026033012250439100_ref419","first-page":"28","article-title":"\u201cMetody ustraneniya neratsional\u2019nykh perevozok pri- planirovanii [Russian; Methods of removing irrational transportation in planning]\u201d","volume":"9","author":"Tolstoi","year":"1939","journal-title":"Sotsialisticheskii Transport"},{"issue":"2","key":"2026033012250439100_ref420","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1007\/s10208-004-0128-z","article-title":"\u201cMetamorphoses through Lie group action\u201d","volume":"5","author":"Trouv\u00e9","year":"2005","journal-title":"Foundations of Computational Mathematics"},{"issue":"1","key":"2026033012250439100_ref421","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1007\/PL00009922","article-title":"\u201cOn the Monge mass transfer problem\u201d","volume":"13","author":"Trudinger","year":"2001","journal-title":"Calculus of Variations and Partial Differential Equations"},{"key":"2026033012250439100_ref422","first-page":"1","volume-title":"Information Processing in Medical Imaging","author":"Vaillant","year":"2005"},{"issue":"2","key":"2026033012250439100_ref423","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1002\/cpa.3160200210","article-title":"\u201cOn the behavior of the fundamental solution of the heat equation with variable coefficients\u201d","volume":"20","author":"Varadhan","year":"1967","journal-title":"Communications on Pure and Applied Mathematics"},{"key":"2026033012250439100_ref424","volume-title":"Graduate Studies in Mathematics Series","author":"Villani","year":"2003"},{"key":"2026033012250439100_ref425","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume":"Vol. 338","author":"Villani","year":"2009","journal-title":"Optimal Transport: Old and New"},{"issue":"9","key":"2026033012250439100_ref426","doi-asserted-by":"crossref","first-page":"1482","DOI":"10.1007\/s10851-018-0827-8","article-title":"\u201cMeasure-valued variational models with applications to diffusion-weighted imaging\u201d","volume":"60","author":"Vogt","year":"2018","journal-title":"Journal of Mathematical Imaging and Vision"},{"key":"2026033012250439100_ref427","first-page":"442","volume-title":"\u201cSupervised earth mover\u2019s distance learning and its computer vision applications\u201d","author":"Wang","year":"2012"},{"issue":"3","key":"2026033012250439100_ref428","doi-asserted-by":"crossref","first-page":"621","DOI":"10.1109\/TMI.2010.2089693","article-title":"\u201cAn optimal transportation approach for nuclear structure-based pathology\u201d","volume":"30","author":"Wang","year":"2011","journal-title":"IEEE Transactions on Medical Imaging"},{"issue":"2","key":"2026033012250439100_ref429","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1007\/s11263-012-0566-z","article-title":"\u201cA linear optimal transportation framework for quantifying and visualizing variations in sets of images\u201d","volume":"101","author":"Wang","year":"2013","journal-title":"International Journal of Computer Vision"},{"key":"2026033012250439100_ref430","volume-title":"\u201cSharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance\u201d","author":"Weed","year":"2017"},{"key":"2026033012250439100_ref431","first-page":"207","article-title":"\u201cDistance metric learning for large margin nearest neighbor classification\u201d","volume":"10","author":"Weinberger","year":"2009","journal-title":"Journal of Machine Learning Research"},{"issue":"1","key":"2026033012250439100_ref432","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1051\/m2an\/2009043","article-title":"\u201cVariational particle schemes for the porous medium equation and for the system of isentropic Euler equations\u201d","volume":"44","author":"Westdickenberg","year":"2010","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"key":"2026033012250439100_ref433","first-page":"108","volume-title":"Journal of Transport Economics and Policy","author":"Wilson","year":"1969"},{"issue":"1","key":"2026033012250439100_ref434","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1137\/130918010","article-title":"\u201cSynthesizing and mixing stationary Gaussian texture models\u201d","volume":"7","author":"Xia","year":"2014","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"6","key":"2026033012250439100_ref435","doi-asserted-by":"crossref","first-page":"579","DOI":"10.2307\/2340126","article-title":"\u201cOn the methods of measuring association between two attributes\u201d","volume":"75","author":"Yule","year":"1912","journal-title":"Journal of the Royal Statistical Society"},{"key":"2026033012250439100_ref436","volume-title":"to appear in Bernoulli","author":"Zemel","year":"2018"},{"key":"2026033012250439100_ref437","first-page":"3690","volume-title":"Proceedings of ICPR\u201914","author":"Zen","year":"2014"},{"issue":"6","key":"2026033012250439100_ref438","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1109\/TIP.2007.896637","article-title":"\u201cAn image morphing technique based on optimal mass preserving mapping\u201d","volume":"16","author":"Zhu","year":"2007","journal-title":"IEEE Transactions on Image Processing"},{"issue":"4","key":"2026033012250439100_ref439","doi-asserted-by":"crossref","first-page":"3397","DOI":"10.1007\/s00526-015-0909-z","article-title":"\u201cTransport distances and geodesic convexity for systems of degenerate diffusion equations\u201d","volume":"54","author":"Zinsl","year":"2015.","journal-title":"Calculus of Variations and Partial Differential Equations"}],"container-title":["Foundations and Trends\u00ae in Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.emerald.com\/ftmal\/article-pdf\/11\/5-6\/355\/11154291\/2200000073en.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/www.emerald.com\/ftmal\/article-pdf\/11\/5-6\/355\/11154291\/2200000073en.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T18:10:53Z","timestamp":1777486253000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.emerald.com\/ftmal\/article\/11\/5-6\/355\/1332396\/Computational-Optimal-Transport-with-Applications"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,12]]},"references-count":438,"journal-issue":{"issue":"5-6","published-print":{"date-parts":[[2019,2,12]]}},"URL":"https:\/\/doi.org\/10.1561\/2200000073","relation":{},"ISSN":["1935-8237","1935-8245"],"issn-type":[{"value":"1935-8237","type":"print"},{"value":"1935-8245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,2,12]]}}}