{"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/low-rank-matrix-recovery-with-scaled","title":"Low-Rank Matrix Recovery with Scaled Subgradient Methods: Fast and Robust Convergence Without the Condition Number","arxiv_id":"2010.13364","date":"2020-10-26","proceeding":null,"authors":["Tian Tong","Cong Ma","Yuejie Chi"],"abstract":"Many problems in data science can be treated as estimating a low-rank matrix from highly incomplete, sometimes even corrupted, observations. One popular approach is to resort to matrix factorization, where the low-rank matrix factors are optimized via first-order methods over a smooth loss function, such as the residual sum of squares. While tremendous progresses have been made in recent years, the natural smooth formulation suffers from two sources of ill-conditioning, where the iteration complexity of gradient descent scales poorly both with the dimension as well as the condition number of the low-rank matrix. Moreover, the smooth formulation is not robust to corruptions. In this paper, we propose scaled subgradient methods to minimize a family of nonsmooth and nonconvex formulations -- in particular, the residual sum of absolute errors -- which is guaranteed to converge at a fast rate that is almost dimension-free and independent of the condition number, even in the presence of corruptions. We illustrate the effectiveness of our approach when the observation operator satisfies certain mixed-norm restricted isometry properties, and derive state-of-the-art performance guarantees for a variety of problems such as robust low-rank matrix sensing and quadratic sampling.","url_abs":"https://arxiv.org/abs/2010.13364v2","url_pdf":"https://arxiv.org/pdf/2010.13364v2.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":"low-rank-matrix-recovery-with-scaled","repo_url":"https://github.com/Titan-Tong/ScaledGD","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"low-rank-matrix-recovery-with-scaled","repo_url":"https://github.com/pilancilab/Riemannian_Preconditioned_LoRA","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2010.13364","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}