{"paper":{"title":"Linear-size $\\ell_1$ sparsifiers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.MG","authors_text":"Thomas Rothvoss, Victor Reis","submitted_at":"2026-06-26T14:43:15Z","abstract_excerpt":"We prove that for any matrix $A \\in \\mathbb{R}^{m \\times n}$ and any $\\varepsilon \\in (0, 1/2]$ there is a diagonal matrix $D \\in \\mathbb{R}_{\\geq 0}^{m \\times m}$ with at most $O(\\frac{n}{\\varepsilon^2} \\log(\\frac{1}{\\varepsilon}))$ nonzero entries so that \\[(1-\\varepsilon) \\|Ax\\|_1 \\leq \\|DAx\\|_1 \\leq (1+\\varepsilon)\\|Ax\\|_1 \\quad \\forall x \\in \\mathbb{R}^n.\\]In particular, for any zonotope $Z \\subseteq \\mathbb{R}^{n}$ there exists a zonotope $Z' \\subseteq \\mathbb{R}^{n}$ generated by at most $O(\\frac{n}{\\varepsilon^2} \\log(\\frac{1}{\\varepsilon}))$ segments so that $(1-\\varepsilon) Z \\subset"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.28147","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.28147/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}