{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T15:27:52Z","timestamp":1787239672456,"version":"build-2736575974"},"reference-count":20,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000879","name":"Alfred P. Sloan Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000879","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2123224"],"award-info":[{"award-number":["DMS-2123224"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Rev."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:p>t-distributed stochastic neighborhood embedding (t-SNE), a clustering and visualization method proposed by van der Maaten and Hinton in 2008, has rapidly become a standard tool in the natural sciences. Despite its overwhelming success, it has a distinct lack of mathematical foundations and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters. As a by-product, the proof suggests that t-SNE is merely one of many possible algorithms of a large family of methods generated by dynamical systems---this perspective suggests new questions and problems, some of which we discuss.<\/jats:p>","DOI":"10.1137\/21m1446769","type":"journal-article","created":{"date-parts":[[2022,2,2]],"date-time":"2022-02-02T16:30:57Z","timestamp":1643819457000},"page":"153-178","source":"Crossref","is-referenced-by-count":8,"title":["Dimensionality Reduction via Dynamical Systems: The Case of t-SNE"],"prefix":"10.1137","volume":"64","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0074-0346","authenticated-orcid":true,"given":"George C.","family":"Linderman","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefan","family":"Steinerberger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,2,3]]},"reference":[{"key":"atypb1","unstructured":"S. Arora, W. Hu, and P. K. Kothari (2018),\n                      An analysis of the t-SNE algorithm for data visualization\n                      , in Conference on Learning Theory (PLMR 2018), pp. 1455-1462."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1162\/089976603321780317"},{"key":"atypb3","unstructured":"J. N. B\u00f6hm, P. Berens, and D. Kobak (2020),\n                      A Unifying Perspective on Neighbor Embeddings along the Attraction-Repulsion Spectrum\n                      , preprint,https:\/\/arxiv.org\/abs\/2007.08902."},{"key":"atypb4","unstructured":"T. T. Cai and R. Ma (2021),\n                      Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data\n                      , preprint,https:\/\/arxiv.org\/abs\/2105.07536."},{"key":"atypb5","unstructured":"M. A. Carreira-Perpinan (2010),\n                      The elastic embedding algorithm for dimensionality reduction\n                      , in 27th Int. Conf. Machine Learning (ICML 2010), pp. 167-174."},{"key":"atypb6","unstructured":"G. Hinton and S. T. Roweis (2002),\n                      Stochastic neighbor embedding\n                      , in Advances in Neural Information Processing Systems (NIPS 15), MIT Press, pp. 833-840."},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"M. Jacomy, T. Venturini, S. Heymann, and M. Bastian (2014),\n                      ForceAtlas$2$, a continuous graph layout algorithm for handy network visualization designed for the Gephi software\n                      , PloS One, 9, art. e98679.","DOI":"10.1371\/journal.pone.0098679"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-019-13056-x"},{"key":"atypb9","doi-asserted-by":"crossref","unstructured":"D. Kobak, G. Linderman, S. Steinerberger, Y. Kluger, and P. Berens (2019),\n                      Heavy-tailed kernels reveal a finer cluster structure in t-SNE visualisations\n                      , in the Joint European Conference on Machine Learning and Knowledge Discovery in Databases, Springer, Cham, pp. 124-139.","DOI":"10.1007\/978-3-030-46150-8_8"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1038\/s41587-020-00809-z"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1038\/s41592-018-0308-4"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/18M1216134"},{"key":"atypb13","first-page":"3221","volume":"15","author":"van der Maaten L.","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"atypb14","first-page":"2579","volume":"9","author":"van der Maaten L.","year":"2008","journal-title":"J. Mach. Learn. Res."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cell.2015.05.002"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"L. McInnes, J. Healy, and J. Melville (2018),\n                      UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction\n                      , preprint,https:\/\/arxiv.org\/abs\/1802.03426.","DOI":"10.21105\/joss.00861"},{"key":"atypb17","unstructured":"I. Robinson and E. Pierce-Hoffman (2020),\n                      Tree-SNE: Hierarchical Clustering and Visualization Using t-SNE\n                      , preprint,https:\/\/arxiv.org\/abs\/2002.05687."},{"key":"atypb18","unstructured":"U. Shaham and S. Steinerberger (2017),\n                      Stochastic Neighbor Embedding Separates Well-Separated Clusters\n                      , preprint,https:\/\/arxiv.org\/abs\/1702.02670."},{"key":"atypb19","first-page":"2169","volume":"22","author":"Yang Z.","year":"2009","journal-title":"Adv. Neural Inform. Process. Syst."},{"key":"atypb20","doi-asserted-by":"crossref","unstructured":"Y. Zhang and S. Steinerberger (2021),\n                      t-SNE, Forceful Colorings and Mean Field Limits\n                      , preprint,https:\/\/arxiv.org\/abs\/2102.13009.","DOI":"10.1007\/s40687-022-00340-4"}],"container-title":["SIAM Review"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/21M1446769","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:44:44Z","timestamp":1787237084000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/21M1446769"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["10.1137\/21M1446769"],"URL":"https:\/\/doi.org\/10.1137\/21m1446769","relation":{},"ISSN":["0036-1445","1095-7200"],"issn-type":[{"value":"0036-1445","type":"print"},{"value":"1095-7200","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,2]]}}}