{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:31:29Z","timestamp":1787333489391,"version":"3.56.0"},"reference-count":34,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1638521"],"award-info":[{"award-number":["DMS-1638521"]}],"id":[{"id":"10.13039\/100000001","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":[[2021,1]]},"abstract":"<jats:p>Probabilistic models of data sets often exhibit salient geometric structure. Such a phenomenon is summed up in the manifold distribution hypothesis and can be exploited in probabilistic learning. Here we present normal-bundle bootstrap (NBB), a method that generates new data which preserve the geometric structure of a given data set. Inspired by algorithms for manifold learning and concepts in differential geometry, our method decomposes the underlying probability measure into a marginalized measure on a learned data manifold and conditional measures on the normal spaces. The algorithm estimates the data manifold as a density ridge and constructs new data by bootstrapping projection vectors and adding them to the ridge. We apply our method to the inference of density ridge and related statistics, and to data augmentation to reduce overfitting.<\/jats:p>","DOI":"10.1137\/20m1356002","type":"journal-article","created":{"date-parts":[[2021,5,4]],"date-time":"2021-05-04T11:45:38Z","timestamp":1620128738000},"page":"573-592","source":"Crossref","is-referenced-by-count":4,"title":["Normal-Bundle Bootstrap"],"prefix":"10.1137","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2899-1008","authenticated-orcid":true,"given":"Ruda","family":"Zhang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1890-920X","authenticated-orcid":true,"given":"Roger","family":"Ghanem","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,4]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/040605266"},{"key":"atypb2","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1109\/TPAMI.2013.50","volume":"35","author":"Bengio Y.","year":"2013","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"atypb3","doi-asserted-by":"crossref","first-page":"2257","DOI":"10.3150\/16-BEJ810","volume":"23","author":"Betancourt M.","year":"2017","journal-title":"Bernoulli"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"A. Bhattacharya and R. Bhattacharya,\n                      Nonparametric Inference on Manifolds: With Applications to Shape Spaces\n                      , Cambridge University Press, Cambridge, UK, 2012,https:\/\/doi.org\/10.1017\/CBO9781139094764.","DOI":"10.1017\/CBO9781139094764"},{"key":"atypb5","unstructured":"C. Bishop,\n                      Pattern Recognition and Machine Learning\n                      , Springer, New York, 2006."},{"key":"atypb6","first-page":"161","author":"Brubaker M.","year":"2012","journal-title":"PMLR"},{"key":"atypb7","doi-asserted-by":"crossref","first-page":"825","DOI":"10.1111\/sjos.12036","volume":"40","author":"Byrne S.","year":"2013","journal-title":"Scand. J. Stat."},{"key":"atypb8","unstructured":"Y.C. Chen,\n                      Solution Manifold and Its Statistical Applications\n                      , preprint,https:\/\/arxiv.org\/abs\/2002.05297, 2020."},{"key":"atypb9","unstructured":"Y.C. Chen, C. R. Genovese, S. Ho, and L. Wasserman,\n                      Optimal ridge detection using coverage risk\n                      , in Advances in Neural Information Processing Systems 28, NeurIPS, San Deigo, CA, 2015, pp. 316-324,http:\/\/papers.nips.cc\/paper\/5996-optimal-ridge-detection-using-coverage-risk."},{"key":"atypb10","first-page":"489","volume":"44","author":"Chen Y.-C.","year":"2016","journal-title":"Ann. Statist."},{"key":"atypb11","first-page":"1896","volume":"43","author":"Chen Y.-C.","year":"2015","journal-title":"Ann. Statist."},{"key":"atypb12","doi-asserted-by":"crossref","unstructured":"Y. Chikuse,\n                      Statistics on Special Manifolds\n                      , Lect. Notes Stat. 174, Springer-Verlag, New York, 2003,https:\/\/doi.org\/10.1007\/978-0-387-21540-2.","DOI":"10.1007\/978-0-387-21540-2"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/0727079"},{"key":"atypb14","first-page":"102","author":"Diaconis P.","year":"2013","journal-title":"OH"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"B. Efron and R. Tibshirani,\n                      An Introduction to the Bootstrap\n                      , Chapman and Hall, New York, 1993.","DOI":"10.1007\/978-1-4899-4541-9"},{"key":"atypb16","doi-asserted-by":"crossref","first-page":"598","DOI":"10.1080\/10618600.2016.1260472","volume":"26","author":"Genovese C.","year":"2017","journal-title":"J. Comput. Graph. Statist."},{"key":"atypb17","doi-asserted-by":"crossref","first-page":"1511","DOI":"10.1214\/14-AOS1218","volume":"42","author":"Genovese C. R.","year":"2014","journal-title":"Ann. Statist."},{"key":"atypb18","unstructured":"I. Goodfellow, Y. Bengio, and A. Courville,\n                      Deep Learning\n                      , MIT Press, Cambridge, MA, 2016,https:\/\/mitpress.mit.edu\/books\/deep-learning."},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"J. Guckenheimer and P. Holmes,\n                      Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields\n                      , Appl. Math. Sci. 42, Springer, New York, 1983,https:\/\/doi.org\/10.1007\/978-1-4612-1140-2.","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"atypb20","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1080\/01621459.1989.10478797","volume":"84","author":"Hastie T.","year":"1989","journal-title":"J. Amer. Statist. Assoc."},{"key":"atypb21","doi-asserted-by":"crossref","unstructured":"M. W. Hirsch,\n                      Differential Topology\n                      , Grad. Texts Math. 33, Springer, New York, NY, 1976,https:\/\/doi.org\/10.1007\/978-1-4684-9449-5.","DOI":"10.1007\/978-1-4684-9449-5"},{"key":"atypb22","unstructured":"M. C. Irwin,\n                      Smooth Dynamical Systems\n                      , Academic Press, New York, 1980."},{"key":"atypb23","doi-asserted-by":"crossref","unstructured":"J. M. Lee,\n                      Introduction to Smooth Manifolds\n                      , Grad. Texts Math. 218, Springer, New York, 2012,https:\/\/doi.org\/10.1007\/978-1-4419-9982-5.","DOI":"10.1007\/978-1-4419-9982-5"},{"key":"atypb24","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s00211-019-01056-4","volume":"143","author":"Lelievre T.","year":"2019","journal-title":"Numer. Math."},{"key":"atypb25","first-page":"1249","volume":"12","author":"Ozertem U.","year":"2011","journal-title":"J. Mach. Learn. Res."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"V. Patrangenaru and L. Ellingson,\n                      Nonparametric Statistics on Manifolds and Their Applications to Object Data Analysis\n                      , CRC Press, Boca Raton, FL, 2015,https:\/\/doi.org\/10.1201\/b18969.","DOI":"10.1201\/b18969"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"G. Rudolph and M. Schmidt,\n                      Differential Geometry and Mathematical Physics. PartI.Manifolds, Lie Groups and Hamiltonian Systems\n                      , Springer, Dordrecht, The Netherlands, 2013,https:\/\/doi.org\/10.1007\/978-94-007-5345-7.","DOI":"10.1007\/978-94-007-5345-7"},{"key":"atypb28","doi-asserted-by":"crossref","first-page":"242","DOI":"10.1016\/j.jcp.2016.05.044","volume":"321","author":"Soize C.","year":"2016","journal-title":"J. Comput. Phys."},{"key":"atypb29","unstructured":"C. Soize and R. Ghanem,\n                      Probabilistic Learning on Manifolds\n                      , preprint,https:\/\/arxiv.org\/abs\/2002.12653, 2020."},{"key":"atypb30","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1146\/annurev-statistics-031017-100045","volume":"5","author":"Wasserman L.","year":"2018","journal-title":"Annu. Rev. Stat. Appl."},{"key":"atypb31","first-page":"1261","volume":"14","author":"Wu C.-F. J.","year":"1986","journal-title":"Ann. Statist."},{"key":"atypb32","doi-asserted-by":"crossref","first-page":"2609","DOI":"10.1002\/cpa.21783","volume":"71","author":"Zappa E.","year":"2018","journal-title":"Comm. Pure Appl. Math."},{"key":"atypb33","unstructured":"R. Zhang,\n                      Newton Retraction as Approximate Geodesics on Submanifolds\n                      , preprint,https:\/\/arxiv.org\/abs\/2006.14751, 2020."},{"key":"atypb34","first-page":"10","volume":"27","author":"Zhang R.","year":"2020","journal-title":"Oxford Research Encyclopedia of Environmental Science"}],"container-title":["SIAM Journal on Mathematics of Data Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1356002","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:04:58Z","timestamp":1787331898000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1356002"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1356002"],"URL":"https:\/\/doi.org\/10.1137\/20m1356002","relation":{},"ISSN":["2577-0187"],"issn-type":[{"value":"2577-0187","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}