{"paper":{"title":"Almost Separable Matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Karen Gunderson, Leonardo Baldassini, Matthew Aldridge","submitted_at":"2014-10-07T17:59:46Z","abstract_excerpt":"An $m \\times n$ matrix $\\mathsf{A}$ with column supports $\\{S_i\\}$ is $k$-separable if the disjunctions $\\bigcup_{i \\in \\mathcal{K}} S_i$ are all distinct over all sets $\\mathcal{K}$ of cardinality $k$. While a simple counting bound shows that $m > k \\log_2 n/k$ rows are required for a separable matrix to exist, in fact it is necessary for $m$ to be about a factor of $k$ more than this. In this paper, we consider a weaker definition of `almost $k$-separability', which requires that the disjunctions are `mostly distinct'. We show using a random construction that these matrices exist with $m = O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1826","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"}