{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T10:53:55Z","timestamp":1777719235169,"version":"3.51.4"},"reference-count":91,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"7","license":[{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T00:00:00Z","timestamp":1719792000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62106271"],"award-info":[{"award-number":["62106271"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101606"],"award-info":[{"award-number":["12101606"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001042"],"award-info":[{"award-number":["12001042"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Renmin University of China research fund program for young scholars"},{"DOI":"10.13039\/501100012236","name":"Beijing Institute of Technology research fund program for young scholars","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100012236","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Beijing Municipal Natural Science Foundation","award":["1232019"],"award-info":[{"award-number":["1232019"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Pattern Anal. Mach. Intell."],"published-print":{"date-parts":[[2024,7]]},"DOI":"10.1109\/tpami.2024.3363780","type":"journal-article","created":{"date-parts":[[2024,2,8]],"date-time":"2024-02-08T18:38:26Z","timestamp":1707417506000},"page":"4993-5007","source":"Crossref","is-referenced-by-count":6,"title":["Hilbert Curve Projection Distance for Distribution Comparison"],"prefix":"10.1109","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6497-6489","authenticated-orcid":false,"given":"Tao","family":"Li","sequence":"first","affiliation":[{"name":"Institute of Statistics and Big Data, Renmin University of China, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7111-0966","authenticated-orcid":false,"given":"Cheng","family":"Meng","sequence":"additional","affiliation":[{"name":"Center for Applied Statistics, Institute of Statistics and Big Data, Renmin University of China, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4192-5360","authenticated-orcid":false,"given":"Hongteng","family":"Xu","sequence":"additional","affiliation":[{"name":"Gaoling School of Artificial Intelligence, Beijing Key Laboratory of Big Data Management and Analysis Methods, Renmin University of China, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6068-8415","authenticated-orcid":false,"given":"Jun","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"263","reference":[{"key":"ref1","first-page":"957","article-title":"From word embeddings to document distances","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Kusner"},{"key":"ref2","first-page":"4869","article-title":"Supervised word movers distance","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Huang"},{"key":"ref3","article-title":"Distance measure machines","author":"Rakotomamonjy","year":"2018"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.5555\/2969033.2969125"},{"key":"ref5","first-page":"1","article-title":"Auto-encoding variational bayes","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Kingma"},{"key":"ref6","first-page":"214","article-title":"Wasserstein generative adversarial networks","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Arjovsky"},{"issue":"1","key":"ref7","first-page":"723","article-title":"A kernel two-sample test","volume":"13","author":"Gretton","year":"2012","journal-title":"J. Mach. Learn. Res."},{"key":"ref8","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal Transport: Old and New","volume":"338","author":"Villani","year":"2009"},{"key":"ref9","first-page":"1","article-title":"Wasserstein auto-encoders","volume-title":"Proc. 6th Int. Conf. Learn. Representations","author":"Tolstikhin"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s002050050044"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1002\/fld.264"},{"key":"ref12","first-page":"661","article-title":"The earth movers distance, multi-dimensional scaling, and color-based image retrieval","volume-title":"Proc. ARPA Image Understanding Workshop","author":"Rubner"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.2009.5459199"},{"key":"ref14","first-page":"2292","article-title":"Sinkhorn distances: Lightspeed computation of optimal transport","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Cuturi"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-014-0506-3"},{"key":"ref16","article-title":"Generalized sliced Wasserstein distances","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Kolouri"},{"key":"ref17","article-title":"Tree-sliced variants of Wasserstein distances","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Le"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1007\/springerreference_62889"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1080\/02693799008941526"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1109\/69.908985"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2019.01090"},{"key":"ref22","first-page":"1","article-title":"Sliced Wasserstein auto-encoders","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Kolouri"},{"key":"ref23","first-page":"1961","article-title":"Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Altschuler"},{"key":"ref24","first-page":"4429","article-title":"Massively scalable sinkhorn distances via the Nystr\u00f6m method","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Altschuler"},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-49988-4_28"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.3390\/a14050143"},{"key":"ref27","first-page":"1","article-title":"Importance sparsification for sinkhorn algorithm","volume":"24","author":"Li","year":"2023","journal-title":"J. Mach. Learn. Res."},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.4310\/CMS.2022.v20.n7.a11"},{"key":"ref29","first-page":"2088","article-title":"Fast algorithms for computational optimal transport and Wasserstein barycenter","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Guo"},{"key":"ref30","first-page":"1367","article-title":"Computational optimal transport: Complexity by accelerated gradient descent is better than by Sinkhorns algorithm","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Dvurechensky"},{"key":"ref31","first-page":"3440","article-title":"Stochastic optimization for large-scale optimal transport","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Genevay"},{"key":"ref32","first-page":"433","article-title":"A fast proximal point method for computing exact Wasserstein distance","volume-title":"Proc. Int. Conf. Uncertainty Artif. Intell.","author":"Xie"},{"key":"ref33","first-page":"2816","article-title":"Bregman alternating direction method of multipliers","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Wang"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2017.2659647"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2022.3153126"},{"key":"ref36","first-page":"1","article-title":"Distributional sliced-Wasserstein and applications to generative modeling","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Nguyen"},{"key":"ref37","first-page":"186","article-title":"Orthogonal estimation of Wasserstein distances","volume-title":"Proc. 22nd Int. Conf. Artif. Intell. Statist.","author":"Rowland"},{"key":"ref38","first-page":"5072","article-title":"Subspace robust Wasserstein distances","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Paty"},{"key":"ref39","first-page":"9383","article-title":"Projection robust Wasserstein distance and Riemannian optimization","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Lin"},{"key":"ref40","first-page":"262","article-title":"On projection robust optimal transport: Sample complexity and model misspecification","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Lin"},{"key":"ref41","article-title":"Revisiting sliced Wasserstein on images: From vectorization to convolution","author":"Nguyen","year":"2022"},{"key":"ref42","article-title":"Amortized projection optimization for sliced Wasserstein generative models","author":"Nguyen","year":"2022"},{"key":"ref43","first-page":"1608","article-title":"Learning generative models with Sinkhorn divergences","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Genevay"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2019.00383"},{"key":"ref45","first-page":"10576","article-title":"Learning autoencoders with relational regularization","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Xu"},{"key":"ref46","first-page":"2053","article-title":"Learning with a Wasserstein loss","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Frogner"},{"key":"ref47","article-title":"Wasserstein Weisfeiler-Lehman graph kernels","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Togninalli"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1080\/10618600.2023.2165500"},{"key":"ref49","first-page":"4015","article-title":"Sufficient dimension reduction for classification using principal optimal transport direction","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Meng"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1007\/s10994-018-5717-1"},{"key":"ref51","first-page":"3733","article-title":"Joint distribution optimal transportation for domain adaptation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Courty"},{"key":"ref52","first-page":"6932","article-title":"Gromov-Wasserstein learning for graph matching and node embedding","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Xu"},{"key":"ref53","article-title":"Scalable Gromov-Wasserstein learning for graph partitioning and matching","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Xu"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1109\/ICIP.2014.7025983"},{"key":"ref55","first-page":"8116","article-title":"Large-scale optimal transport map estimation using projection pursuit","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Meng"},{"key":"ref56","article-title":"Hierarchical optimal transport for document representation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Yurochkin"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12132"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-322-80063-3"},{"key":"ref59","first-page":"2131","article-title":"Learning with minibatch Wasserstein: Asymptotic and gradient properties","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Fatras"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1007\/s00205-013-0669-x"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.3150\/18-BEJ1065"},{"key":"ref62","first-page":"13961","article-title":"Distributional convergence of the sliced Wasserstein process","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Xi"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-statistics-030718-104938"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1561\/2200000073"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(69)80010-3"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1109\/T-C.1971.223258"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1063\/1.1751381"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-46759-7_20"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1145\/1653771.1653865"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2007.08.034"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1111\/rssb.12312"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2015.7298801"},{"key":"ref73","first-page":"652","article-title":"PointNet: Deep learning on point sets for 3D classification and segmentation","volume-title":"Proc. IEEE Conf. Comput. Vis. Pattern Recognit.","author":"Qi"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.48550\/arXiv.1310.4546"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1109\/5.726791"},{"key":"ref76","first-page":"1","article-title":"Unsupervised representation learning with deep convolutional generative adversarial networks","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Radford"},{"key":"ref77","first-page":"1","article-title":"Adam: A method for stochastic optimization","volume-title":"Proc. Int. Conf. Learn. Representations","author":"Kingma"},{"key":"ref78","first-page":"6629","article-title":"GANs trained by a two time-scale update rule converge to a local Nash equilibrium","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Heusel"},{"key":"ref79","first-page":"12934","article-title":"Robust optimal transport with applications in generative modeling and domain adaptation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Balaji"},{"key":"ref80","first-page":"7850","article-title":"Outlier-robust optimal transport","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Mukherjee"},{"key":"ref81","first-page":"21947","article-title":"On robust optimal transport: Computational complexity and barycenter computation","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Le"},{"key":"ref82","first-page":"7673","article-title":"On unbalanced optimal transport: An analysis of sinkhorn algorithm","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Pham"},{"key":"ref83","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3303"},{"key":"ref84","article-title":"Unbalanced optimal transport meets sliced-Wasserstein","author":"S\u00e9journ\u00e9","year":"2023"},{"key":"ref85","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-011-9093-5"},{"key":"ref86","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2015020"},{"key":"ref87","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2021.3077465"},{"key":"ref88","doi-asserted-by":"publisher","DOI":"10.1137\/100805741"},{"key":"ref89","first-page":"2664","article-title":"Gromov-Wasserstein averaging of kernel and distance matrices","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Peyr\u00e9"},{"key":"ref90","doi-asserted-by":"publisher","DOI":"10.1002\/cpe.1179"},{"key":"ref91","first-page":"500","article-title":"Hilbert R-tree: An improved R-tree using fractals","volume-title":"Proc. 20th Int. Conf. Very Large Data Bases","author":"Kamel","year":"1994"}],"container-title":["IEEE Transactions on Pattern Analysis and Machine Intelligence"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/34\/10550108\/10428036.pdf?arnumber=10428036","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,6,25]],"date-time":"2024-06-25T19:57:55Z","timestamp":1719345475000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/10428036\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7]]},"references-count":91,"journal-issue":{"issue":"7"},"URL":"https:\/\/doi.org\/10.1109\/tpami.2024.3363780","relation":{},"ISSN":["0162-8828","2160-9292","1939-3539"],"issn-type":[{"value":"0162-8828","type":"print"},{"value":"2160-9292","type":"electronic"},{"value":"1939-3539","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7]]}}}