{"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/constructive-subsampling-of-finite-frames","title":"Constructive subsampling of finite frames with applications in optimal function recovery","arxiv_id":"2202.12625","date":"2022-02-25","proceeding":null,"authors":["Felix Bartel","Martin Schäfer","Tino Ullrich"],"abstract":"In this paper we present new constructive methods, random and deterministic, for the efficient subsampling of finite frames in $\\mathbb C^m$. Based on a suitable random subsampling strategy, we are able to extract from any given frame with bounds $0<A\\le B<\\infty$ (and condition $B/A$) a similarly conditioned reweighted subframe consisting of merely $\\mathcal{O}(m\\log m)$ elements. Further, utilizing a deterministic subsampling method based on principles developed by Batson, Spielman, and Srivastava to control the spectrum of sums of Hermitian rank-1 matrices, we are able to reduce the number of elements to $\\mathcal{O}(m)$ (with a constant close to one). By controlling the weights via a preconditioning step, we can, in addition, preserve the lower frame bound in the unweighted case. This permits the derivation of new quasi-optimal unweighted (left) Marcinkiewicz-Zygmund inequalities for $L_2(D,\\nu)$ with constructible node sets of size $\\mathcal{O}(m)$ for $m$-dimensional subspaces of bounded functions. Those can be applied e.g. for (plain) least-squares sampling reconstruction of functions, where we obtain new quasi-optimal results avoiding the Kadison-Singer theorem. Numerical experiments indicate the applicability of our results.","url_abs":"https://arxiv.org/abs/2202.12625v3","url_pdf":"https://arxiv.org/pdf/2202.12625v3.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":"constructive-subsampling-of-finite-frames","repo_url":"https://github.com/felixbartel/bsssubsampling.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}