{"paper":{"title":"Improved Lower Bounds for the Restricted Isometry Property of Subsampled Fourier Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","math.IT","math.PR"],"primary_cat":"cs.IT","authors_text":"Shravas Rao","submitted_at":"2019-03-28T17:34:10Z","abstract_excerpt":"Let $A$ be an $N \\times N$ Fourier matrix over $\\mathbb{F}_p^{\\log{N}/\\log{p}}$ for some prime $p$. We improve upon known lower bounds for the number of rows of $A$ that must be sampled so that the resulting matrix $M$ satisfies the restricted isometry property for $k$-sparse vectors. This property states that $\\|Mv\\|_2^2$ is approximately $\\|v\\|_2^2$ for all $k$-sparse vectors $v$. In particular, if $k = \\Omega( \\log^2{N})$, we show that $\\Omega(k\\log{k}\\log{N}/\\log{p})$ rows must be sampled to satisfy the restricted isometry property with constant probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.12146","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":""},"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"}