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In the projective clustering problem, given $k, q$ and norm $\\rho \\in [1,\\infty]$, we have to compute a set $\\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\\sum_{p\\in P}d(p, \\mathcal{F})^\\rho)^{1/\\rho}$ is minimized; here $d(p, \\mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\\mathcal{F}$. We let $f_k^q(P,\\rho)$ denote the minimal value and interpret $f_k^q(P,\\infty)$ to be $\\max_{r\\in P}d(r, \\mathcal{F})$. When $\\rho=1,2$ and $\\infty$ and $q=0$, the problem corresponds to the $k$-median, $k$-mean and the "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1407.2063","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CG","submitted_at":"2014-07-08T12:34:11Z","cross_cats_sorted":[],"title_canon_sha256":"591e5b147d6e5e5c86b110a3deaccf6392b291e5c5e65f43620288fa38570dad","abstract_canon_sha256":"2bcbe0b4d64909a7dfcd5ccce1b03f88c11a4a6f94473fde501af87f103a05e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:59:53.589204Z","signature_b64":"8aFFYsFYfRPAbRZxkpieLVQq6+zWIxHUB0dR7KP17PUfNnUpcS37+Zz3+ceAv+h0npm3Cs0X7c0JOJjNAYcMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34980c782ce66444e8d85545e34f346bedf6867008313ec1903e6f1c7b3c3d76","last_reissued_at":"2026-05-18T01:59:53.588690Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:59:53.588690Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Approximation and Streaming Algorithms for Projective Clustering via Random Projections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Michael Kerber, Sharath Raghvendra","submitted_at":"2014-07-08T12:34:11Z","abstract_excerpt":"Let $P$ be a set of $n$ points in $\\mathbb{R}^d$. In the projective clustering problem, given $k, q$ and norm $\\rho \\in [1,\\infty]$, we have to compute a set $\\mathcal{F}$ of $k$ $q$-dimensional flats such that $(\\sum_{p\\in P}d(p, \\mathcal{F})^\\rho)^{1/\\rho}$ is minimized; here $d(p, \\mathcal{F})$ represents the (Euclidean) distance of $p$ to the closest flat in $\\mathcal{F}$. We let $f_k^q(P,\\rho)$ denote the minimal value and interpret $f_k^q(P,\\infty)$ to be $\\max_{r\\in P}d(r, \\mathcal{F})$. 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