{"paper":{"title":"Small coherence implies the weak Null Space Property","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"St\\'ephane Chr\\'etien, Zhen Wai Olivier Ho","submitted_at":"2016-06-29T17:29:05Z","abstract_excerpt":"In the Compressed Sensing community, it is well known that given a matrix $X \\in \\mathbb R^{n\\times p}$ with $\\ell_2$ normalized columns, the Restricted Isometry Property (RIP) implies the Null Space Property (NSP). It is also well known that a small Coherence $\\mu$ implies a weak RIP, i.e. the singular values of $X_T$ lie between $1-\\delta$ and $1+\\delta$ for \"most\" index subsets $T \\subset \\{1,\\ldots,p\\}$ with size governed by $\\mu$ and $\\delta$. In this short note, we show that a small Coherence implies a weak Null Space Property, i.e. $\\Vert h_T\\Vert_2 \\le C \\ \\Vert h_{T^c}\\Vert_1/\\sqrt{s}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.09193","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"}