{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T02:58:21Z","timestamp":1787885901719,"version":"build-2784847793"},"reference-count":7,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"name":"Institute of New Economic Thinking","award":["INO15-00038"],"award-info":[{"award-number":["INO15-00038"]}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["1R01HG008383-01A1"],"award-info":[{"award-number":["1R01HG008383-01A1"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["T32GM007205"],"award-info":[{"award-number":["T32GM007205"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM Journal on Mathematics of Data Science"],"published-print":{"date-parts":[[2019,1]]},"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 a number of natural sciences. Despite its overwhelming success, there is 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; more precisely, we prove that t-SNE in the \u201cearly exaggeration\u201d phase, an optimization technique proposed by van der Maaten and Hinton [ J. Mach. Learn. Res., 9 (2008), pp. 2579--2605] and van der Maaten [ J. Mach. Learn. Res., 15 (2014), pp. 3221--3245], can be rigorously analyzed. As a byproduct, the proof suggests novel ways for setting the exaggeration parameter $\\alpha$ and step size $h$. Numerical examples illustrate the effectiveness of these rules: in particular, the quality of embedding of topological structures (e.g., the swiss roll) improves. We also discuss a connection to spectral clustering methods.<\/jats:p>","DOI":"10.1137\/18m1216134","type":"journal-article","created":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T13:08:40Z","timestamp":1559048920000},"page":"313-332","source":"Crossref","is-referenced-by-count":217,"title":["Clustering with t-SNE, Provably"],"prefix":"10.1137","volume":"1","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":[[2019,5,28]]},"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                      , preprint,arXiv:1803.01768."},{"key":"atypb2","doi-asserted-by":"crossref","first-page":"1373","DOI":"10.1162\/089976603321780317","volume":"15","author":"Belkin M.","year":"2003","journal-title":"Neural Comput."},{"key":"atypb3","first-page":"167","author":"Carreira-Perpinan M. A.","year":"2010","journal-title":"Int. Conf. Machine Learning"},{"key":"atypb4","first-page":"3221","volume":"15","author":"van der Maaten L.","year":"2014","journal-title":"J. Mach. Learn. Res."},{"key":"atypb5","first-page":"2579","volume":"9","author":"van der Maaten L.","year":"2008","journal-title":"J. Mach. Learn. Res."},{"key":"atypb6","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1016\/j.cell.2015.05.002","volume":"161","author":"Macosko E. Z.","year":"2015","journal-title":"Cell"},{"key":"atypb7","unstructured":"U. Shaham and S. Steinerberger (2017),\n                      Stochastic Neighbor Embedding Separates Well-Separated Cluster\n                      , preprint,arXiv:1702.02670."}],"container-title":["SIAM Journal on Mathematics of Data Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1216134","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:40:24Z","timestamp":1787330424000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1216134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":7,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1216134"],"URL":"https:\/\/doi.org\/10.1137\/18m1216134","relation":{},"ISSN":["2577-0187"],"issn-type":[{"value":"2577-0187","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}