{"paper":{"title":"The Restricted Isometry Property of Subsampled Fourier Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT","math.PR"],"primary_cat":"cs.DS","authors_text":"Ishay Haviv, Oded Regev","submitted_at":"2015-07-07T11:59:52Z","abstract_excerpt":"A matrix $A \\in \\mathbb{C}^{q \\times N}$ satisfies the restricted isometry property of order $k$ with constant $\\varepsilon$ if it preserves the $\\ell_2$ norm of all $k$-sparse vectors up to a factor of $1\\pm \\varepsilon$. We prove that a matrix $A$ obtained by randomly sampling $q = O(k \\cdot \\log^2 k \\cdot \\log N)$ rows from an $N \\times N$ Fourier matrix satisfies the restricted isometry property of order $k$ with a fixed $\\varepsilon$ with high probability. This improves on Rudelson and Vershynin (Comm. Pure Appl. Math., 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.01768","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}