{"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/sharpness-and-well-conditioning-of-nonsmooth","title":"Sharpness and well-conditioning of nonsmooth convex formulations in statistical signal recovery","arxiv_id":"2307.06873","date":"2023-07-13","proceeding":null,"authors":["Lijun Ding","Alex L. Wang"],"abstract":"We study a sample complexity vs. conditioning tradeoff in modern signal recovery problems (including sparse recovery, low-rank matrix sensing, covariance estimation, and abstract phase retrieval), where convex optimization problems are built from sampled observations. We begin by introducing a set of condition numbers related to sharpness in $\\ell_p$ or Schatten-$p$ norms ($p\\in[1,2]$) of a nonsmooth formulation for these problems. Then, we show that these condition numbers become dimension independent constants in each of the example signal recovery problems once the sample size exceeds some constant multiple of the recovery threshold. Structurally, this result ensures that the inaccuracy in the recovered signal due to both observation noise and optimization error is well-controlled. Algorithmically, such a result ensures that a new restarted mirror descent method achieves nearly-dimension-independent linear convergence to the signal. This new first-order method is general and applies to any sharp convex function in an $\\ell_p$ or Schatten-$p$ norm ($p\\in[1,2]$).","url_abs":"https://arxiv.org/abs/2307.06873v2","url_pdf":"https://arxiv.org/pdf/2307.06873v2.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":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"sharpness-and-well-conditioning-of-nonsmooth","repo_url":"https://github.com/alexlihengwang/sharpness_well_conditioning","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}