{"paper":{"title":"Approximation Algorithms for Clustering via Weighted Impurity Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Eduardo Laber, Ferdinando Cicalese","submitted_at":"2018-07-13T18:14:11Z","abstract_excerpt":"An impurity measures $I:{R}^k \\to {R}^+$ maps a $k$-dimensional vector ${\\bf v}$ to a non-negative value $I({\\bf v})$ so that the more homogeneous ${\\bf v}$, the larger its impurity. We study clustering based on impurity measures: given a collection $V$ of $n$ many $k$-dimensional vectors and an impurity measure $I$, the goal is to find a partition ${\\cal P}$ of $V$ into $L$ groups $V_1,\\ldots,V_L$ that minimizes the total impurities of the groups in ${\\cal P}$, i.e., $I({\\cal P})= \\sum_{m=1}^{L} I(\\sum_{{\\bf v} \\in V_m}{\\bf v}).$\n  Impurity minimization is widely used as quality assessment me"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.05241","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"}