{"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/unique-sparse-decomposition-of-low-rank","title":"Unique sparse decomposition of low rank matrices","arxiv_id":"2106.07736","date":"2021-06-14","proceeding":"NeurIPS 2021 12","authors":["Dian Jin","Xin Bing","Yuqian Zhang"],"abstract":"The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix $Y\\in \\mathbb{R}^{p\\times n}$ that admits a sparse representation. Specifically, we consider $Y = A X\\in \\mathbb{R}^{p\\times n}$ where the matrix $A\\in \\mathbb{R}^{p\\times r}$ has full column rank, with $r < \\min\\{n,p\\}$, and the matrix $X\\in \\mathbb{R}^{r\\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. At last, we corroborate these theoretical results with numerical experiments.","url_abs":"https://arxiv.org/abs/2106.07736v5","url_pdf":"https://arxiv.org/pdf/2106.07736v5.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":"unique-sparse-decomposition-of-low-rank","repo_url":"https://github.com/Jindiande/Unique_Fac_of_Low_Rank","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2106.07736","atlas_url":"https://app.syntology.ai/?focus=2106.07736","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}