{"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/fixed-sparsity-matrix-approximation-from","title":"Fixed-sparsity matrix approximation from matrix-vector products","arxiv_id":"2402.09379","date":"2024-02-14","proceeding":null,"authors":["Noah Amsel","Tyler Chen","Feyza Duman Keles","Diana Halikias","Cameron Musco","Christopher Musco"],"abstract":"We study the problem of approximating a matrix $\\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized algorithm that returns an approximation with the given sparsity pattern with Frobenius-norm error at most $(1+\\varepsilon)$ times the best possible error. When each row of the desired sparsity pattern has at most $s$ nonzero entries, this algorithm requires $O(s/\\varepsilon)$ non-adaptive matrix-vector products with $\\mathbf{A}$. We also prove a matching lower-bound, showing that, for any sparsity pattern with $\\Theta(s)$ nonzeros per row and column, any algorithm achieving $(1+\\epsilon)$ approximation requires $\\Omega(s/\\varepsilon)$ matrix-vector products in the worst case. We thus resolve the matrix-vector product query complexity of the problem up to constant factors, even for the well-studied case of diagonal approximation, for which no previous lower bounds were known.","url_abs":"https://arxiv.org/abs/2402.09379v3","url_pdf":"https://arxiv.org/pdf/2402.09379v3.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":"fixed-sparsity-matrix-approximation-from","repo_url":"https://github.com/tchen-research/fixed_sparsity_matrix_approximation","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2402.09379","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}