{"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/sparse-subspace-clustering-algorithm-theory","title":"Sparse Subspace Clustering: Algorithm, Theory, and Applications","arxiv_id":"1203.1005","date":"2012-03-05","proceeding":null,"authors":["Ehsan Elhamifar","Rene Vidal"],"abstract":"In many real-world problems, we are dealing with collections of\nhigh-dimensional data, such as images, videos, text and web documents, DNA\nmicroarray data, and more. Often, high-dimensional data lie close to\nlow-dimensional structures corresponding to several classes or categories the\ndata belongs to. In this paper, we propose and study an algorithm, called\nSparse Subspace Clustering (SSC), to cluster data points that lie in a union of\nlow-dimensional subspaces. The key idea is that, among infinitely many possible\nrepresentations of a data point in terms of other points, a sparse\nrepresentation corresponds to selecting a few points from the same subspace.\nThis motivates solving a sparse optimization program whose solution is used in\na spectral clustering framework to infer the clustering of data into subspaces.\nSince solving the sparse optimization program is in general NP-hard, we\nconsider a convex relaxation and show that, under appropriate conditions on the\narrangement of subspaces and the distribution of data, the proposed\nminimization program succeeds in recovering the desired sparse representations.\nThe proposed algorithm can be solved efficiently and can handle data points\nnear the intersections of subspaces. Another key advantage of the proposed\nalgorithm with respect to the state of the art is that it can deal with data\nnuisances, such as noise, sparse outlying entries, and missing entries,\ndirectly by incorporating the model of the data into the sparse optimization\nprogram. We demonstrate the effectiveness of the proposed algorithm through\nexperiments on synthetic data as well as the two real-world problems of motion\nsegmentation and face clustering.","url_abs":"http://arxiv.org/abs/1203.1005v3","url_pdf":"http://arxiv.org/pdf/1203.1005v3.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":"sparse-subspace-clustering-algorithm-theory","repo_url":"https://github.com/IIT-PAVIS/subspace-clustering-action-recognition","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"NOASSERTION"}},{"paper_slug":"sparse-subspace-clustering-algorithm-theory","repo_url":"https://github.com/panji1990/Deep-subspace-clustering-networks","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"unanswered"}},{"paper_slug":"sparse-subspace-clustering-algorithm-theory","repo_url":"https://github.com/panji530/Deep-subspace-clustering-networks","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok"}},{"paper_slug":"sparse-subspace-clustering-algorithm-theory","repo_url":"https://github.com/sohanghosh29/Clustering-Codes","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"clustering","task_name":"Clustering"},{"task_slug":"face-clustering","task_name":"Face Clustering"},{"task_slug":"image-clustering","task_name":"Image Clustering"},{"task_slug":"motion-segmentation","task_name":"Motion Segmentation"}],"methods":[{"method_slug":"spectral-clustering","method_name":"Spectral Clustering"}],"datasets_introduced":[],"methods_introduced":[],"results":[{"leaderboard":"/sota/image-clustering-on-extended-yale-b","task":"Image Clustering","dataset":"Extended Yale-B","model":"SSC","rank_in_archive_order":7,"of":9,"metrics":{"Accuracy":"0.706"},"uses_additional_data":false},{"leaderboard":"/sota/motion-segmentation-on-hopkins155","task":"Motion Segmentation","dataset":"Hopkins155","model":"SSC","rank_in_archive_order":4,"of":4,"metrics":{"Classification Error":"2.18"},"uses_additional_data":false}],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1203.1005","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}