{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:SX7CG2YSZCTXPKZPPR2S6SXODO","short_pith_number":"pith:SX7CG2YS","schema_version":"1.0","canonical_sha256":"95fe236b12c8a777ab2f7c752f4aee1b9ee0eec0a497783c6110bfbdfb8cf783","source":{"kind":"arxiv","id":"1607.06203","version":1},"attestation_state":"computed","paper":{"title":"Greedy bi-criteria approximations for $k$-medians and $k$-means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.DS","authors_text":"Daniel Hsu, Matus Telgarsky","submitted_at":"2016-07-21T06:04:36Z","abstract_excerpt":"This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of $k$-medians and $k$-means, the key results are as follows.\n  $\\bullet$ When the method considers all data points as candidate centers, then selecting $\\mathcal{O}(k\\log(1/\\varepsilon))$ centers achieves cost at most $2+\\varepsilon$ times the optimal cost with $k$ centers.\n  $\\bullet$ Alternatively, the same guarantees hold if each round samples $"},"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":"1607.06203","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2016-07-21T06:04:36Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"82f0ae45f507a60813ef6877d3f8611c9e4b3405190477e2cc98866bdba51436","abstract_canon_sha256":"19aaacb1f7a2d5a3145008379bfd5d429047fdcddc3bbf0afebf2fd2c16d0ba2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:42.274454Z","signature_b64":"6Em3kJGDo7ZQqkWaynSs0c3D7lItrlLWuear/dPgE9QRTSCae5ngIzuwuo2U6lTtfJ9NMq0Q5E4gVfHcBdq1AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"95fe236b12c8a777ab2f7c752f4aee1b9ee0eec0a497783c6110bfbdfb8cf783","last_reissued_at":"2026-05-18T01:10:42.273762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:42.273762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Greedy bi-criteria approximations for $k$-medians and $k$-means","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.DS","authors_text":"Daniel Hsu, Matus Telgarsky","submitted_at":"2016-07-21T06:04:36Z","abstract_excerpt":"This paper investigates the following natural greedy procedure for clustering in the bi-criterion setting: iteratively grow a set of centers, in each round adding the center from a candidate set that maximally decreases clustering cost. In the case of $k$-medians and $k$-means, the key results are as follows.\n  $\\bullet$ When the method considers all data points as candidate centers, then selecting $\\mathcal{O}(k\\log(1/\\varepsilon))$ centers achieves cost at most $2+\\varepsilon$ times the optimal cost with $k$ centers.\n  $\\bullet$ Alternatively, the same guarantees hold if each round samples $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06203","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1607.06203","created_at":"2026-05-18T01:10:42.273876+00:00"},{"alias_kind":"arxiv_version","alias_value":"1607.06203v1","created_at":"2026-05-18T01:10:42.273876+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.06203","created_at":"2026-05-18T01:10:42.273876+00:00"},{"alias_kind":"pith_short_12","alias_value":"SX7CG2YSZCTX","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_16","alias_value":"SX7CG2YSZCTXPKZP","created_at":"2026-05-18T12:30:44.179134+00:00"},{"alias_kind":"pith_short_8","alias_value":"SX7CG2YS","created_at":"2026-05-18T12:30:44.179134+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO","json":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO.json","graph_json":"https://pith.science/api/pith-number/SX7CG2YSZCTXPKZPPR2S6SXODO/graph.json","events_json":"https://pith.science/api/pith-number/SX7CG2YSZCTXPKZPPR2S6SXODO/events.json","paper":"https://pith.science/paper/SX7CG2YS"},"agent_actions":{"view_html":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO","download_json":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO.json","view_paper":"https://pith.science/paper/SX7CG2YS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1607.06203&json=true","fetch_graph":"https://pith.science/api/pith-number/SX7CG2YSZCTXPKZPPR2S6SXODO/graph.json","fetch_events":"https://pith.science/api/pith-number/SX7CG2YSZCTXPKZPPR2S6SXODO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO/action/storage_attestation","attest_author":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO/action/author_attestation","sign_citation":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO/action/citation_signature","submit_replication":"https://pith.science/pith/SX7CG2YSZCTXPKZPPR2S6SXODO/action/replication_record"}},"created_at":"2026-05-18T01:10:42.273876+00:00","updated_at":"2026-05-18T01:10:42.273876+00:00"}