{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:6G5YGVO4LP5SBZAXGNWCTUWGRX","short_pith_number":"pith:6G5YGVO4","schema_version":"1.0","canonical_sha256":"f1bb8355dc5bfb20e417336c29d2c68ddf7dcd1573d91a0f50ed5aadbeb69bf4","source":{"kind":"arxiv","id":"1706.08996","version":3},"attestation_state":"computed","paper":{"title":"Good Clusterings Have Large Volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"F. Happach, S. Borgwardt","submitted_at":"2017-06-27T18:20:40Z","abstract_excerpt":"The clustering of a data set is one of the core tasks in data analytics. Many clustering algorithms exhibit a strong contrast between a favorable performance in practice and bad theoretical worst-cases. Prime examples are least-squares assignments and the popular $k$-means algorithm. We are interested in this contrast and study it through polyhedral theory. Several popular clustering algorithms can be connected to finding a vertex of the so-called bounded-shape partition polytopes. The vertices correspond to clusterings with extraordinary separation properties, in particular allowing the const"},"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":"1706.08996","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2017-06-27T18:20:40Z","cross_cats_sorted":[],"title_canon_sha256":"8455f7e6cf40936c97d5f80ae9d6c54b9aea8f6ada5a6ea597ae642a29ac7325","abstract_canon_sha256":"8e53d403f72c6ba052ac24d92352f6379245d994d85ec11e6edb1c6f4f94867d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:41.748629Z","signature_b64":"fIXDPffPQAV9rCmkFqCt9glPDKIeohvRk/pvncJ16tATFQt7CPETi4dPr3ir/ZOy5BWmEWv0vwEef050Tq0DAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f1bb8355dc5bfb20e417336c29d2c68ddf7dcd1573d91a0f50ed5aadbeb69bf4","last_reissued_at":"2026-05-18T00:06:41.748088Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:41.748088Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Good Clusterings Have Large Volume","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"F. Happach, S. Borgwardt","submitted_at":"2017-06-27T18:20:40Z","abstract_excerpt":"The clustering of a data set is one of the core tasks in data analytics. Many clustering algorithms exhibit a strong contrast between a favorable performance in practice and bad theoretical worst-cases. Prime examples are least-squares assignments and the popular $k$-means algorithm. We are interested in this contrast and study it through polyhedral theory. Several popular clustering algorithms can be connected to finding a vertex of the so-called bounded-shape partition polytopes. The vertices correspond to clusterings with extraordinary separation properties, in particular allowing the const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.08996","kind":"arxiv","version":3},"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":"1706.08996","created_at":"2026-05-18T00:06:41.748170+00:00"},{"alias_kind":"arxiv_version","alias_value":"1706.08996v3","created_at":"2026-05-18T00:06:41.748170+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.08996","created_at":"2026-05-18T00:06:41.748170+00:00"},{"alias_kind":"pith_short_12","alias_value":"6G5YGVO4LP5S","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"6G5YGVO4LP5SBZAX","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"6G5YGVO4","created_at":"2026-05-18T12:31:03.183658+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/6G5YGVO4LP5SBZAXGNWCTUWGRX","json":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX.json","graph_json":"https://pith.science/api/pith-number/6G5YGVO4LP5SBZAXGNWCTUWGRX/graph.json","events_json":"https://pith.science/api/pith-number/6G5YGVO4LP5SBZAXGNWCTUWGRX/events.json","paper":"https://pith.science/paper/6G5YGVO4"},"agent_actions":{"view_html":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX","download_json":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX.json","view_paper":"https://pith.science/paper/6G5YGVO4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1706.08996&json=true","fetch_graph":"https://pith.science/api/pith-number/6G5YGVO4LP5SBZAXGNWCTUWGRX/graph.json","fetch_events":"https://pith.science/api/pith-number/6G5YGVO4LP5SBZAXGNWCTUWGRX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX/action/storage_attestation","attest_author":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX/action/author_attestation","sign_citation":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX/action/citation_signature","submit_replication":"https://pith.science/pith/6G5YGVO4LP5SBZAXGNWCTUWGRX/action/replication_record"}},"created_at":"2026-05-18T00:06:41.748170+00:00","updated_at":"2026-05-18T00:06:41.748170+00:00"}