{"id":1138391,"date":"2025-05-01T09:30:24","date_gmt":"2025-05-01T16:30:24","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=1138391"},"modified":"2025-05-01T09:30:25","modified_gmt":"2025-05-01T16:30:25","slug":"guessing-efficiently-for-constrained-subspace-approximation","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/guessing-efficiently-for-constrained-subspace-approximation\/","title":{"rendered":"Guessing Efficiently for Constrained Subspace Approximation"},"content":{"rendered":"

In this paper we study constrained subspace approximation problem. Given a set of points {a_1,…,a_n} in R^d , the goal of the subspace approximation<\/em> problem is to find a k-dimensional subspace that best approximates the input points. More precisely, for a given p>=1, we aim to minimize the pth power of the l_p norm of the error vector (||a_1- P<\/strong>a_1||,…,||a_n-P<\/strong>a_n||), where P<\/strong> denotes the projection matrix onto the subspace and the norms are Euclidean. In constrained<\/em> subspace approximation (CSA), we additionally have constraints on the projection matrix P<\/strong>. In its most general form, we require P<\/strong> to belong to a given subset S that is described explicitly or implicitly.
\nWe introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either (1+\u03b5)<\/span><\/span><\/span><\/span>-multiplicative or \u03b5<\/span><\/span><\/span><\/span>-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to fair<\/em> subspace approximation, k-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for k-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.<\/p>\n","protected":false},"excerpt":{"rendered":"

In this paper we study constrained subspace approximation problem. Given a set of points {a_1,…,a_n} in R^d , the goal of the subspace approximation problem is to find a k-dimensional subspace that best approximates the input points. More precisely, for a given p>=1, we aim to minimize the pth power of the l_p norm of […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"","msr_pages_string":"","msr_page_range_start":"","msr_page_range_end":"","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"International Colloquium on Automata, Languages, and Programming 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