{"paper":{"title":"On the Separability of Stochastic Geometric Objects, with Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Jie Xue, Ravi Janardan, Yuan Li","submitted_at":"2016-03-22T22:51:55Z","abstract_excerpt":"In this paper, we study the linear separability problem for stochastic geometric objects under the well-known unipoint/multipoint uncertainty models. Let $S=S_R \\cup S_B$ be a given set of stochastic bichromatic points, and define $n = \\min\\{|S_R|, |S_B|\\}$ and $N = \\max\\{|S_R|, |S_B|\\}$. We show that the separable-probability (SP) of $S$ can be computed in $O(nN^{d-1})$ time for $d \\geq 3$ and $O(\\min\\{nN \\log N, N^2\\})$ time for $d=2$, while the expected separation-margin (ESM) of $S$ can be computed in $O(nN^{d})$ time for $d \\geq 2$. In addition, we give an $\\Omega(nN^{d-1})$ witness-based"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.07021","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"}