{"id":190,"date":"2013-05-18T22:37:57","date_gmt":"2013-05-18T22:37:57","guid":{"rendered":"http:\/\/web.eecs.umich.edu\/~girasole\/wordpress\/?page_id=190"},"modified":"2026-07-23T01:47:44","modified_gmt":"2026-07-23T01:47:44","slug":"subspace-tracking","status":"publish","type":"page","link":"https:\/\/web.eecs.umich.edu\/~girasole\/?page_id=190","title":{"rendered":"Subspace Tracking"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">Data in the real world often have a great deal of structure. One way to capture that structure is with principal components or singular vectors. If one is interested in the best k vectors to approximate a dataset, the top k singular vectors provide exactly that. These vectors span the best-fit subspace to the data. I study the estimation of these subspaces as well as algorithms to track subspaces that change over time. I&#8217;m interested in understanding the impact of singular value gaps, noise, and corruption on subspace estimation and tracking.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For code, see posts on <a href=\"http:\/\/web.eecs.umich.edu\/~girasole\/?p=110\">GROUSE<\/a>, an l2 subspace tracking algorithm, <a href=\"http:\/\/web.eecs.umich.edu\/~girasole\/?p=113\">GRASTA<\/a>, an l1 subspace tracking algorithm, its Open CV version <a href=\"http:\/\/web.eecs.umich.edu\/~girasole\/?p=115\">GRASTAcam<\/a>, and <a href=\"http:\/\/web.eecs.umich.edu\/~girasole\/?p=315\">TGRASTA<\/a>, an algorithm that estimates a subspace under non-linear transformations.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>L. Balzano, \u201c<strong>On the Equivalence of Oja\u2019s Algorithm and GROUSE<\/strong>.\u201d Proceedings of Artificial Intelligence and Statistics (AI Stats), March 2022.<\/li>\n\n\n\n<li>K. Gilman, D.A. Tarzanagh, and L. Balzano, \u201c<strong>Grassmannian Optimization for Online Tensor Completion and Tracking with the t-SVD<\/strong>.\u201d <em>IEEE Transactions on Signal Processing,<\/em> vol. 70 (2022): 2152-2167.<\/li>\n\n\n\n<li>H. Lyu, D. Needell, and L. Balzano, \u201c<strong>Online Matrix Factorization for Markovian Data and Applications to Network Dictionary Learning<\/strong>.\u201d <em>Journal of Machine Learning Research, <\/em>21, no. 251 (2020): 1-49.<\/li>\n\n\n\n<li>L. Balzano, Y. Chi, and Y. Lu, &#8220;<strong>Streaming PCA and subspace tracking: The missing data case.<\/strong>&#8221; Proceedings of the IEEE 106, no. 8 (2018): 1293-1310.<\/li>\n\n\n\n<li>D. Zhang and L. Balzano, \u201c<strong>Global Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation.<\/strong>\u201d Proceedings of Artificial Intelligence and Statistics (AI Stats, poster presentation 31% acceptance), May 2016.<\/li>\n\n\n\n<li>L. Balzano, R. Nowak, and B. Recht, \u201c<strong>Online and Adaptive Tracking of Subspaces from Highly Incomplete Information<\/strong>,\u201d Proceedings of the Allerton Conference on Communications, Control, and Computing, September 2010.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Data in the real world often have a great deal of structure. One way to capture that structure is with principal components or singular vectors. If one is interested in the best k vectors to approximate a dataset, the top k singular vectors provide exactly that. These vectors span the best-fit subspace to the data. [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"parent":182,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-190","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/pages\/190","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=190"}],"version-history":[{"count":11,"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/pages\/190\/revisions"}],"predecessor-version":[{"id":1079,"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/pages\/190\/revisions\/1079"}],"up":[{"embeddable":true,"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=\/wp\/v2\/pages\/182"}],"wp:attachment":[{"href":"https:\/\/web.eecs.umich.edu\/~girasole\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=190"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}