{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,20]],"date-time":"2025-11-20T12:43:09Z","timestamp":1763642589011},"reference-count":56,"publisher":"MIT Press","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Neural Computation"],"published-print":{"date-parts":[[2019,4]]},"abstract":"<jats:p>The need to reason about uncertainty in large, complex, and multimodal data sets has become increasingly common across modern scientific environments. The ability to transform samples from one distribution [Formula: see text] to another distribution [Formula: see text] enables the solution to many problems in machine learning (e.g., Bayesian inference, generative modeling) and has been actively pursued from theoretical, computational, and application perspectives across the fields of information theory, computer science, and biology. Performing such transformations in general still leads to computational difficulties, especially in high dimensions. Here, we consider the problem of computing such \u201cmeasure transport maps\u201d with efficient and parallelizable methods. Under the mild assumptions that [Formula: see text] need not be known but can be sampled from and that the density of [Formula: see text] is known up to a proportionality constant, and that [Formula: see text] is log-concave, we provide in this work a convex optimization problem pertaining to relative entropy minimization. We show how an empirical minimization formulation and polynomial chaos map parameterization can allow for learning a transport map between [Formula: see text] and [Formula: see text] with distributed and scalable methods. We also leverage findings from nonequilibrium thermodynamics to represent the transport map as a composition of simpler maps, each of which is learned sequentially with a transport cost regularized version of the aforementioned problem formulation. We provide examples of our framework within the context of Bayesian inference for the Boston housing data set and generative modeling for handwritten digit images from the MNIST data set.<\/jats:p>","DOI":"10.1162\/neco_a_01172","type":"journal-article","created":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T02:01:44Z","timestamp":1550196104000},"page":"613-652","source":"Crossref","is-referenced-by-count":8,"title":["A Distributed Framework for the Construction of Transport Maps"],"prefix":"10.1162","volume":"31","author":[{"given":"Diego A.","family":"Mesa","sequence":"first","affiliation":[{"name":"Department of Electrical Engineering and Computer Science and Department of Biomedical Informatics, Vanderbilt University, Nashville, TN 37205, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Justin","family":"Tantiongloc","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Marcela","family":"Mendoza","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, University of California, San Diego, La Jolla, CA 92093, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sanggyun","family":"Kim","sequence":"additional","affiliation":[{"name":"Department of Psychiatry and Behavioral Sciences, Stanford University, Stanford, CA 94304, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Todd","family":"P. Coleman","sequence":"additional","affiliation":[{"name":"Department of Bioengineering, University of California, San Diego, La Jolla, CA 92093, U.S.A."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","reference":[{"key":"B1","doi-asserted-by":"publisher","DOI":"10.1023\/A:1020281327116"},{"key":"B2","author":"Arjovsky M.","year":"2017","journal-title":"Wasserstein GAN"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1007\/s00199-004-0514-4"},{"key":"B5","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0075847"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1137\/141000439"},{"key":"B7","first-page":"1","volume":"54","author":"Benamou J.-D.","year":"2015","journal-title":"ESAIM"},{"key":"B8","author":"Bernardo J. M.","year":"2001","journal-title":"Bayesian theory"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1214\/10-AOP592"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1137\/120874850"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1561\/2200000016"},{"key":"B13","first-page":"805","volume":"305","author":"Brenier Y.","year":"1987","journal-title":"CR Acad. Sci. Paris S\u00e9r. I Math."},{"key":"B14","author":"Claici S.","year":"2018","journal-title":"Stochastic Wasserstein barycenters"},{"key":"B15","first-page":"685","author":"Cuturi M.","year":"2014","journal-title":"Proceedings of the International Conference on Machine Learning"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.07.022"},{"key":"B17","author":"Gelman A.","year":"2014","journal-title":"bayesian data analysis"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.1984.4767596"},{"key":"B19","first-page":"3440","volume-title":"Advances in neural information processing systems","volume":"29","author":"Genevay A.","year":"2016"},{"key":"B20","author":"Genevay A.","year":"2017","journal-title":"GAN and VAE from an optimal transport point of view"},{"key":"B21","author":"Gilbert A. C.","year":"2017","journal-title":"Towards understanding the invertibility of convolutional neural networks"},{"key":"B22","author":"Gilks W. R.","year":"2005","journal-title":"Markov chain Monte Carlo"},{"key":"B23","first-page":"2672","volume-title":"Advances in neural information processing systems","author":"Goodfellow I.","year":"2014"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asp047"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1016\/0095-0696(78)90006-2"},{"key":"B26","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/57.1.97"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(97)00093-6"},{"key":"B28","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141096303359"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141096303359"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1109\/ISIT.2013.6620628"},{"key":"B31","author":"Kim S.","year":"2015","journal-title":"Tractable fully Bayesian inference via convex optimization and optimal transport theory"},{"key":"B32","author":"Kingma D. P.","year":"2013","journal-title":"Auto-encoding variational Bayes"},{"key":"B33","first-page":"29","volume":"15","author":"Larochelle H.","year":"2011","journal-title":"Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics"},{"key":"B34","doi-asserted-by":"publisher","DOI":"10.1109\/5.726791"},{"key":"B35","first-page":"1718","author":"Li Y.","year":"2015","journal-title":"Proceedings of the 32nd International Conference on Machine Learning"},{"key":"B38","author":"Ma R.","year":"2011","journal-title":"Proceedings of the 49th Annual Allerton Conference on Communication, Control, and Computing"},{"key":"B39","author":"Ma R.","year":"2014","journal-title":"Proceedings of the IEEE International Symposium on Information Theory"},{"key":"B40","author":"Marzouk Y. M.","year":"2016","journal-title":"An introduction to sampling via measure transport"},{"key":"B41","doi-asserted-by":"publisher","DOI":"10.1109\/ISIT.2015.7282540"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.1137\/130920058"},{"key":"B43","doi-asserted-by":"publisher","DOI":"10.1198\/016214508000000337"},{"key":"B44","author":"Parno M.","year":"2014","journal-title":"Transport map accelerated Markov chain Monte Carlo"},{"key":"B45","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032478"},{"key":"B46","first-page":"1530","author":"Rezende D. J.","year":"2015","journal-title":"Proceedings of the 32nd International Conference on Machine Learning"},{"key":"B47","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4145-2"},{"key":"B48","author":"Salimans T.","year":"2018","journal-title":"Improving gans using optimal transport"},{"key":"B49","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20828-2"},{"key":"B50","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1170-9","author":"Schoutens W.","year":"2000","journal-title":"Stochastic processes and orthogonal polynomials"},{"key":"B51","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198568315.001.0001","author":"Sivia D.","year":"2006","journal-title":"Data analysis: A Bayesian tutorial"},{"key":"B52","author":"Spantini A.","year":"2016","journal-title":"Proceedings of the 30th Conference on Neural Information Processing Systems"},{"key":"B53","author":"Srivastava S.","year":"2015","journal-title":"Scalable Bayes via barycenter in Wasserstein space"},{"key":"B54","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2016.2608723"},{"key":"B55","author":"Tolstikhin I.","year":"2017","journal-title":"Wasserstein auto-encoders"},{"key":"B56","author":"Villani C.","year":"2003","journal-title":"Topics in optimal transportation"},{"key":"B57","author":"Villani C.","year":"2008","journal-title":"Optimal transport: Old and new"},{"key":"B58","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0892-3"},{"key":"B59","first-page":"46","volume":"32","author":"Zhong W.","year":"2014","journal-title":"Journal of Machine Learning Research"},{"key":"B60","doi-asserted-by":"publisher","DOI":"10.1214\/009053607000000127"}],"container-title":["Neural Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mitpressjournals.org\/doi\/pdf\/10.1162\/neco_a_01172","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,14]],"date-time":"2024-07-14T20:16:22Z","timestamp":1720988182000},"score":1,"resource":{"primary":{"URL":"https:\/\/direct.mit.edu\/neco\/article\/31\/4\/613-652\/8464"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4]]},"references-count":56,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["10.1162\/neco_a_01172"],"URL":"https:\/\/doi.org\/10.1162\/neco_a_01172","relation":{},"ISSN":["0899-7667","1530-888X"],"issn-type":[{"value":"0899-7667","type":"print"},{"value":"1530-888X","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4]]}}}