{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:2TPC4TQZWAY2URUT72KB4TVQXB","short_pith_number":"pith:2TPC4TQZ","schema_version":"1.0","canonical_sha256":"d4de2e4e19b031aa4693fe941e4eb0b85185eaafad62ce9198050be1cda0b3bd","source":{"kind":"arxiv","id":"2506.06990","version":2},"attestation_state":"computed","paper":{"title":"Modified K-means Algorithm with Local Optimality Guarantees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Akiko Takeda, Michael R. Metel, Mingyi Li","submitted_at":"2025-06-08T04:37:28Z","abstract_excerpt":"The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm"},"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":"2506.06990","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2025-06-08T04:37:28Z","cross_cats_sorted":["math.OC","stat.ML"],"title_canon_sha256":"fd6b1b9ceffe9a7924bc10e453db2f9c7919bdd1c94035d738d1a32c7cb71021","abstract_canon_sha256":"3b487e0ffc50ca45b58c116d0a1d696aeba28198543f3904b5d79ded3d67da82"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:19:28.654167Z","signature_b64":"iFd9oK7HU2ZkeB3AoCfVvZbn7wCpVYd8NkrVQQJoac6H9RKNK6vts3eSllc1gsKXQgFXG3Jq4gRF4i6bG1zFBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d4de2e4e19b031aa4693fe941e4eb0b85185eaafad62ce9198050be1cda0b3bd","last_reissued_at":"2026-07-05T11:19:28.653608Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:19:28.653608Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modified K-means Algorithm with Local Optimality Guarantees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC","stat.ML"],"primary_cat":"cs.LG","authors_text":"Akiko Takeda, Michael R. Metel, Mingyi Li","submitted_at":"2025-06-08T04:37:28Z","abstract_excerpt":"The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.06990","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2506.06990/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2506.06990","created_at":"2026-07-05T11:19:28.653683+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.06990v2","created_at":"2026-07-05T11:19:28.653683+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.06990","created_at":"2026-07-05T11:19:28.653683+00:00"},{"alias_kind":"pith_short_12","alias_value":"2TPC4TQZWAY2","created_at":"2026-07-05T11:19:28.653683+00:00"},{"alias_kind":"pith_short_16","alias_value":"2TPC4TQZWAY2URUT","created_at":"2026-07-05T11:19:28.653683+00:00"},{"alias_kind":"pith_short_8","alias_value":"2TPC4TQZ","created_at":"2026-07-05T11:19:28.653683+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/2TPC4TQZWAY2URUT72KB4TVQXB","json":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB.json","graph_json":"https://pith.science/api/pith-number/2TPC4TQZWAY2URUT72KB4TVQXB/graph.json","events_json":"https://pith.science/api/pith-number/2TPC4TQZWAY2URUT72KB4TVQXB/events.json","paper":"https://pith.science/paper/2TPC4TQZ"},"agent_actions":{"view_html":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB","download_json":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB.json","view_paper":"https://pith.science/paper/2TPC4TQZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.06990&json=true","fetch_graph":"https://pith.science/api/pith-number/2TPC4TQZWAY2URUT72KB4TVQXB/graph.json","fetch_events":"https://pith.science/api/pith-number/2TPC4TQZWAY2URUT72KB4TVQXB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB/action/storage_attestation","attest_author":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB/action/author_attestation","sign_citation":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB/action/citation_signature","submit_replication":"https://pith.science/pith/2TPC4TQZWAY2URUT72KB4TVQXB/action/replication_record"}},"created_at":"2026-07-05T11:19:28.653683+00:00","updated_at":"2026-07-05T11:19:28.653683+00:00"}