{"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/complete-dictionary-recovery-over-the-sphere-1","title":"Complete Dictionary Recovery over the Sphere","arxiv_id":"1504.06785","date":"2015-04-26","proceeding":null,"authors":["Ju Sun","Qing Qu","John Wright"],"abstract":"We consider the problem of recovering a complete (i.e., square and\ninvertible) matrix $\\mathbf A_0$, from $\\mathbf Y \\in \\mathbb R^{n \\times p}$\nwith $\\mathbf Y = \\mathbf A_0 \\mathbf X_0$, provided $\\mathbf X_0$ is\nsufficiently sparse. This recovery problem is central to the theoretical\nunderstanding of dictionary learning, which seeks a sparse representation for a\ncollection of input signals, and finds numerous applications in modern signal\nprocessing and machine learning. We give the first efficient algorithm that\nprovably recovers $\\mathbf A_0$ when $\\mathbf X_0$ has $O(n)$ nonzeros per\ncolumn, under suitable probability model for $\\mathbf X_0$. In contrast, prior\nresults based on efficient algorithms provide recovery guarantees when $\\mathbf\nX_0$ has only $O(n^{1-\\delta})$ nonzeros per column for any constant $\\delta\n\\in (0, 1)$.\n  Our algorithmic pipeline centers around solving a certain nonconvex\noptimization problem with a spherical constraint, and hence is naturally\nphrased in the language of manifold optimization. To show this apparently hard\nproblem is tractable, we first provide a geometric characterization of the\nhigh-dimensional objective landscape, which shows that with high probability\nthere are no \"spurious\" local minima. This particular geometric structure\nallows us to design a Riemannian trust region algorithm over the sphere that\nprovably converges to one local minimizer with an arbitrary initialization,\ndespite the presence of saddle points. The geometric approach we develop here\nmay also shed light on other problems arising from nonconvex recovery of\nstructured signals.","url_abs":"http://arxiv.org/abs/1504.06785v3","url_pdf":"http://arxiv.org/pdf/1504.06785v3.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":"complete-dictionary-recovery-over-the-sphere-1","repo_url":"https://github.com/sunju/dl_focm","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"dictionary-learning","task_name":"Dictionary Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}