{"paper":{"title":"Random Projections for $k$-means Clustering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.AI","authors_text":"Anastasios Zouzias, Christos Boutsidis, Petros Drineas","submitted_at":"2010-11-21T02:37:10Z","abstract_excerpt":"This paper discusses the topic of dimensionality reduction for $k$-means clustering. We prove that any set of $n$ points in $d$ dimensions (rows in a matrix $A \\in \\RR^{n \\times d}$) can be projected into $t = \\Omega(k / \\eps^2)$ dimensions, for any $\\eps \\in (0,1/3)$, in $O(n d \\lceil \\eps^{-2} k/ \\log(d) \\rceil )$ time, such that with constant probability the optimal $k$-partition of the point set is preserved within a factor of $2+\\eps$. The projection is done by post-multiplying $A$ with a $d \\times t$ random matrix $R$ having entries $+1/\\sqrt{t}$ or $-1/\\sqrt{t}$ with equal probability. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.4632","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"}