{"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/finding-a-sparse-vector-in-a-subspace-linear","title":"Finding a sparse vector in a subspace: Linear sparsity using alternating directions","arxiv_id":"1412.4659","date":"2014-12-15","proceeding":"NeurIPS 2014 12","authors":["Qing Qu","Ju Sun","John Wright"],"abstract":"Is it possible to find the sparsest vector (direction) in a generic subspace\n$\\mathcal{S} \\subseteq \\mathbb{R}^p$ with $\\mathrm{dim}(\\mathcal{S})= n < p$?\nThis problem can be considered a homogeneous variant of the sparse recovery\nproblem, and finds connections to sparse dictionary learning, sparse PCA, and\nmany other problems in signal processing and machine learning. In this paper,\nwe focus on a **planted sparse model** for the subspace: the target sparse\nvector is embedded in an otherwise random subspace. Simple convex heuristics\nfor this planted recovery problem provably break down when the fraction of\nnonzero entries in the target sparse vector substantially exceeds\n$O(1/\\sqrt{n})$. In contrast, we exhibit a relatively simple nonconvex approach\nbased on alternating directions, which provably succeeds even when the fraction\nof nonzero entries is $\\Omega(1)$. To the best of our knowledge, this is the\nfirst practical algorithm to achieve linear scaling under the planted sparse\nmodel. Empirically, our proposed algorithm also succeeds in more challenging\ndata models, e.g., sparse dictionary learning.","url_abs":"http://arxiv.org/abs/1412.4659v3","url_pdf":"http://arxiv.org/pdf/1412.4659v3.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":"finding-a-sparse-vector-in-a-subspace-linear","repo_url":"https://github.com/sunju/psv","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"dictionary-learning","task_name":"Dictionary Learning"}],"methods":[{"method_slug":"pca","method_name":"PCA"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1412.4659","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}