{"paper":{"title":"High Dimensional Clustering with $r$-nets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Alain Ryser, Georgia Avarikioti, Roger Wattenhofer, Yuyi Wang","submitted_at":"2018-11-06T11:13:09Z","abstract_excerpt":"Clustering, a fundamental task in data science and machine learning, groups a set of objects in such a way that objects in the same cluster are closer to each other than to those in other clusters. In this paper, we consider a well-known structure, so-called $r$-nets, which rigorously captures the properties of clustering. We devise algorithms that improve the run-time of approximating $r$-nets in high-dimensional spaces with $\\ell_1$ and $\\ell_2$ metrics from $\\tilde{O}(dn^{2-\\Theta(\\sqrt{\\epsilon})})$ to $\\tilde{O}(dn + n^{2-\\alpha})$, where $\\alpha = \\Omega({\\epsilon^{1/3}}/{\\log(1/\\epsilon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.02288","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"}