{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,4]],"date-time":"2026-07-04T16:57:53Z","timestamp":1783184273051,"version":"3.54.6"},"reference-count":50,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","license":[{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,3,1]],"date-time":"2026-03-01T00:00:00Z","timestamp":1772323200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Signal Process. Mag."],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1109\/msp.2025.3609527","type":"journal-article","created":{"date-parts":[[2026,4,13]],"date-time":"2026-04-13T19:38:42Z","timestamp":1776109122000},"page":"37-50","source":"Crossref","is-referenced-by-count":2,"title":["Flow-Based Generative Models as Iterative Algorithms in Probability Space: An intuitive mathematical framework [Special Issue on the Mathematics of Deep Learning]"],"prefix":"10.1109","volume":"43","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6777-2951","authenticated-orcid":false,"given":"Yao","family":"Xie","sequence":"first","affiliation":[{"name":"H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1034-6019","authenticated-orcid":false,"given":"Xiuyuan","family":"Cheng","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Duke University, Durham, NC, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"263","reference":[{"key":"ref1","article-title":"Building normalizing flows with stochastic interpolants","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Albergo","year":"2023"},{"key":"ref2","first-page":"1","article-title":"Optimizing functionals on the space of probabilities with input convex neural networks","author":"Alvarez-Melis","year":"2022","journal-title":"Trans. Mach. Learn. Res."},{"key":"ref3","volume-title":"Gradient Flows: In Metric Spaces and in the Space of Probability Measures","author":"Ambrosio","year":"2005"},{"key":"ref4","first-page":"3462","article-title":"D-flow: Differentiating through flows for controlled generation","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Ben-Hamu","year":"2024"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050002"},{"key":"ref6","article-title":"Error bounds for flow matching methods","author":"Benton","year":"2024","journal-title":"Trans. Mach. Learn. Res."},{"key":"ref7","first-page":"6572","article-title":"Neural ordinary differential equations","volume-title":"Proc. 32nd Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Chen","year":"2018"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.52202\/075280-2998"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1109\/tit.2024.3422412"},{"key":"ref10","first-page":"2552","article-title":"Density ratio estimation via infinitesimal classification","volume-title":"Proc. Int. Nat. Conf. Artif. Intell. Statist.","author":"Choi","year":"2022"},{"key":"ref11","article-title":"Diffusion posterior sampling for general noisy inverse problems","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Chung","year":"2023"},{"key":"ref12","article-title":"NICE: Non-linear independent components estimation","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR) Workshop","author":"Dinh","year":"2015"},{"key":"ref13","article-title":"Density estimation using real NVP","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Dinh","year":"2017"},{"key":"ref14","first-page":"112,459","article-title":"Stochastic optimal control matching","volume-title":"Proc. 38th Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Domingo I Enrich","year":"2024"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1090\/mbk\/082"},{"key":"ref16","first-page":"6185","article-title":"Variational Wasserstein gradient flow","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Fan","year":"2022"},{"key":"ref17","article-title":"FFJORD: Free-form continuous dynamics for scalable reversible generative models","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Grathwohl","year":"2018"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.90"},{"key":"ref19","first-page":"6840","article-title":"Denoising diffusion probabilistic models","volume-title":"Proc. 34th Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Ho","year":"2020"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/s0036141096303359"},{"key":"ref21","article-title":"Glow: Generative flow with invertible 1x1 convolutions","volume-title":"Proc. 32nd Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Kingma","year":"2018"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1109\/tpami.2020.2992934"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1287\/educ.2019.0198"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2014.34.1533"},{"key":"ref25","article-title":"Towards non-asymptotic convergence for diffusion-based generative models","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Li","year":"2024"},{"key":"ref26","article-title":"Flow matching for generative modeling","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Lipman","year":"2023"},{"key":"ref27","article-title":"Flow matching guide and code","author":"Lipman","year":"2024"},{"key":"ref28","article-title":"Rectified flow: A marginal preserving approach to optimal transport","author":"Liu","year":"2022"},{"issue":"232","key":"ref29","first-page":"1","article-title":"Distribution learning via neural differential equations: A nonparametric statistical perspective","volume":"25","author":"Marzouk","year":"2024","journal-title":"J. Mach. Learn. Res."},{"key":"ref30","first-page":"15,243","article-title":"Large-scale Wasserstein gradient flows","volume-title":"Proc. 35th Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Mokrov","year":"2021"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1109\/ICVGIP.2008.47"},{"key":"ref32","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"Oksendal","year":"2013"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR52734.2025.02638"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/tit.2016.2549542"},{"key":"ref35","first-page":"4905","article-title":"Telescoping density-ratio estimation","volume-title":"Advances in Neural Information Processing Systems","volume":"33","author":"Rhodes","year":"2020"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1922204117"},{"key":"ref37","first-page":"12,356","article-title":"The Wasserstein proximal gradient algorithm","volume-title":"Proc. 34th Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Salim","year":"2020"},{"key":"ref38","article-title":"Progressive distillation for fast sampling of diffusion models","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Salimans","year":"2022"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.2991\/978-94-6239-021-8"},{"key":"ref40","article-title":"Loss-guided diffusion models for plug-and-play controllable generation","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Song","year":"2023"},{"key":"ref41","article-title":"Consistency models","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Song","year":"2023"},{"key":"ref42","article-title":"Score-based generative modeling through stochastic differential equations","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Song","year":"2021"},{"key":"ref43","article-title":"Score-based generative modeling through stochastic differential equations","volume-title":"Proc. Int. Conf. Learn. Representations (ICLR)","author":"Song","year":"2021"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1038\/s41598-023-31521-y"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71050-9"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1109\/jsait.2022.3221864"},{"key":"ref47","first-page":"47,379","article-title":"Normalizing flow neural networks by JKO scheme","volume-title":"Proc. 37th Int. Conf. Neural Inf. Process. Syst. (NIPS)","author":"Xu","year":"2023"},{"key":"ref48","article-title":"Local flow matching generative models","author":"Xu","year":"2024"},{"key":"ref49","article-title":"Computing high-dimensional optimal transport by flow neural networks","volume-title":"Proc. Int. Nat. Conf. Artif. Intell. Statist. (AISTATS)","author":"Xu","year":"2025"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1109\/jsait.2024.3370699"}],"container-title":["IEEE Signal Processing Magazine"],"original-title":[],"link":[{"URL":"https:\/\/ieeexplore.ieee.org\/ielam\/79\/11479943\/11480037-aam.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/xplorestaging.ieee.org\/ielx8\/79\/11479943\/11480037.pdf?arnumber=11480037","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,8]],"date-time":"2026-06-08T19:54:30Z","timestamp":1780948470000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/11480037\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3]]},"references-count":50,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.1109\/msp.2025.3609527","relation":{},"ISSN":["1053-5888","1558-0792"],"issn-type":[{"value":"1053-5888","type":"print"},{"value":"1558-0792","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3]]}}}