{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/nearly-optimal-robust-subspace-tracking","title":"Nearly Optimal Robust Subspace Tracking","arxiv_id":"1712.06061","date":"2017-12-17","proceeding":"ICML 2018 7","authors":["Praneeth Narayanamurthy","Namrata Vaswani"],"abstract":"In this work, we study the robust subspace tracking (RST) problem and obtain\none of the first two provable guarantees for it. The goal of RST is to track\nsequentially arriving data vectors that lie in a slowly changing\nlow-dimensional subspace, while being robust to corruption by additive sparse\noutliers. It can also be interpreted as a dynamic (time-varying) extension of\nrobust PCA (RPCA), with the minor difference that RST also requires a short\ntracking delay. We develop a recursive projected compressive sensing algorithm\nthat we call Nearly Optimal RST via ReProCS (ReProCS-NORST) because its\ntracking delay is nearly optimal. We prove that NORST solves both the RST and\nthe dynamic RPCA problems under weakened standard RPCA assumptions, two simple\nextra assumptions (slow subspace change and most outlier magnitudes lower\nbounded), and a few minor assumptions.\n  Our guarantee shows that NORST enjoys a near optimal tracking delay of $O(r\n\\log n \\log(1/\\epsilon))$. Its required delay between subspace change times is\nthe same, and its memory complexity is $n$ times this value. Thus both these\nare also nearly optimal. Here $n$ is the ambient space dimension, $r$ is the\nsubspaces' dimension, and $\\epsilon$ is the tracking accuracy. NORST also has\nthe best outlier tolerance compared with all previous RPCA or RST methods, both\ntheoretically and empirically (including for real videos), without requiring\nany model on how the outlier support is generated. This is possible because of\nthe extra assumptions it uses.","url_abs":"http://arxiv.org/abs/1712.06061v4","url_pdf":"http://arxiv.org/pdf/1712.06061v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"nearly-optimal-robust-subspace-tracking","repo_url":"https://github.com/praneethmurthy/NORST","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"compressive-sensing","task_name":"Compressive Sensing"}],"methods":[{"method_slug":"pca","method_name":"PCA"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}