{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:RRF6MJA6L6MIO5ERL6R45RXKVF","short_pith_number":"pith:RRF6MJA6","canonical_record":{"source":{"id":"1509.07755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-25T15:30:18Z","cross_cats_sorted":[],"title_canon_sha256":"5786e5762ace97a242f01cfbfbe64303139b7a2ad50e311d9e88b1a29b2ff338","abstract_canon_sha256":"2c2e10ebcc04743ffb9faf3eac4942a0a2cd5c033bb40f7353156efeaea4e9d2"},"schema_version":"1.0"},"canonical_sha256":"8c4be6241e5f988774915fa3cec6eaa979289d427509e8626d4b2e418fbc94a0","source":{"kind":"arxiv","id":"1509.07755","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07755","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07755v1","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07755","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"RRF6MJA6L6MI","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RRF6MJA6L6MIO5ER","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RRF6MJA6","created_at":"2026-05-18T12:29:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:RRF6MJA6L6MIO5ERL6R45RXKVF","target":"record","payload":{"canonical_record":{"source":{"id":"1509.07755","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-25T15:30:18Z","cross_cats_sorted":[],"title_canon_sha256":"5786e5762ace97a242f01cfbfbe64303139b7a2ad50e311d9e88b1a29b2ff338","abstract_canon_sha256":"2c2e10ebcc04743ffb9faf3eac4942a0a2cd5c033bb40f7353156efeaea4e9d2"},"schema_version":"1.0"},"canonical_sha256":"8c4be6241e5f988774915fa3cec6eaa979289d427509e8626d4b2e418fbc94a0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:22.138026Z","signature_b64":"1QORNe2N6OTEVlmNJzFnU/ZOmEuTuGRPs5CMV6ZA7RKnVpfSFE0cemKg654DvesVZqrsC4drfgvLGOv6r2rXBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c4be6241e5f988774915fa3cec6eaa979289d427509e8626d4b2e418fbc94a0","last_reissued_at":"2026-05-18T01:20:22.137283Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:22.137283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1509.07755","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:20:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"msX1ObdwOkIUr5k6Uor95lVpBdEIsQgHMHJvlDPn0SPf5vvSgdOKONKis8fmpm6LTsLyjsW0czyLOHezzvH+Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:22:07.529640Z"},"content_sha256":"81251a70239f4de0d3cf619a92fba5de5ac6305fbee732a8aaab6f6797076067","schema_version":"1.0","event_id":"sha256:81251a70239f4de0d3cf619a92fba5de5ac6305fbee732a8aaab6f6797076067"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:RRF6MJA6L6MIO5ERL6R45RXKVF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Mathematical Theory for Clustering in Metric Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Cheng-Shang Chang, Li-Heng Liou, Wanjiun Liao, Yu-Sheng Chen","submitted_at":"2015-09-25T15:30:18Z","abstract_excerpt":"Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question:\n  What is a cluster in a set of data points?\n  In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07755","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:20:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aZFmRR3Wp/nq8IOxxiFDciFil877BSc6OUjhJMeMcfHgoq6bHYxo3omB5ukPATooWGDiTGQ4iepTaK/wosA3AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T22:22:07.529987Z"},"content_sha256":"1ba9b4253e4b3f641787c7d77d5f799c38a88083fce8d64c17b90f385de5a49e","schema_version":"1.0","event_id":"sha256:1ba9b4253e4b3f641787c7d77d5f799c38a88083fce8d64c17b90f385de5a49e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/bundle.json","state_url":"https://pith.science/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T22:22:07Z","links":{"resolver":"https://pith.science/pith/RRF6MJA6L6MIO5ERL6R45RXKVF","bundle":"https://pith.science/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/bundle.json","state":"https://pith.science/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RRF6MJA6L6MIO5ERL6R45RXKVF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:RRF6MJA6L6MIO5ERL6R45RXKVF","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2c2e10ebcc04743ffb9faf3eac4942a0a2cd5c033bb40f7353156efeaea4e9d2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-25T15:30:18Z","title_canon_sha256":"5786e5762ace97a242f01cfbfbe64303139b7a2ad50e311d9e88b1a29b2ff338"},"schema_version":"1.0","source":{"id":"1509.07755","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1509.07755","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1509.07755v1","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.07755","created_at":"2026-05-18T01:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"RRF6MJA6L6MI","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"RRF6MJA6L6MIO5ER","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"RRF6MJA6","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:1ba9b4253e4b3f641787c7d77d5f799c38a88083fce8d64c17b90f385de5a49e","target":"graph","created_at":"2026-05-18T01:20:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question:\n  What is a cluster in a set of data points?\n  In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms ","authors_text":"Cheng-Shang Chang, Li-Heng Liou, Wanjiun Liao, Yu-Sheng Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-25T15:30:18Z","title":"A Mathematical Theory for Clustering in Metric Spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.07755","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:81251a70239f4de0d3cf619a92fba5de5ac6305fbee732a8aaab6f6797076067","target":"record","created_at":"2026-05-18T01:20:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2c2e10ebcc04743ffb9faf3eac4942a0a2cd5c033bb40f7353156efeaea4e9d2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2015-09-25T15:30:18Z","title_canon_sha256":"5786e5762ace97a242f01cfbfbe64303139b7a2ad50e311d9e88b1a29b2ff338"},"schema_version":"1.0","source":{"id":"1509.07755","kind":"arxiv","version":1}},"canonical_sha256":"8c4be6241e5f988774915fa3cec6eaa979289d427509e8626d4b2e418fbc94a0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c4be6241e5f988774915fa3cec6eaa979289d427509e8626d4b2e418fbc94a0","first_computed_at":"2026-05-18T01:20:22.137283Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:22.137283Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1QORNe2N6OTEVlmNJzFnU/ZOmEuTuGRPs5CMV6ZA7RKnVpfSFE0cemKg654DvesVZqrsC4drfgvLGOv6r2rXBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:22.138026Z","signed_message":"canonical_sha256_bytes"},"source_id":"1509.07755","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81251a70239f4de0d3cf619a92fba5de5ac6305fbee732a8aaab6f6797076067","sha256:1ba9b4253e4b3f641787c7d77d5f799c38a88083fce8d64c17b90f385de5a49e"],"state_sha256":"051d9cfff20c50dfbe83159b6f971bfb2b5a13c3cc3aed5575f65df852597885"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UYQcK4dZRNcunM3QdHQuajWyC1OG/5oidy37YR4GAKQpHagOUQhA0tCx6RgYZacFCLqeFpiT9RD1NFff4XUFDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T22:22:07.531897Z","bundle_sha256":"0e971876fc17a374bf44858623a02be99760cc53a756ddc61226014030d43364"}}