{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:ZUB2YOHYY5L3E7B7GTBGYPTZW7","short_pith_number":"pith:ZUB2YOHY","canonical_record":{"source":{"id":"1812.04356","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-12-11T12:35:36Z","cross_cats_sorted":["stat.ML","stat.TH"],"title_canon_sha256":"3f0301542331eb2230c8583e70867514d0ee4ae7bd569fb22fcc2f85e5d2863d","abstract_canon_sha256":"53b3185cfe134b86dd19d46b31c56c80a9f636b6d16f52b3068dbc7ab8af8835"},"schema_version":"1.0"},"canonical_sha256":"cd03ac38f8c757b27c3f34c26c3e79b7d0dc125363991b80f03d8d0720ddecfd","source":{"kind":"arxiv","id":"1812.04356","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04356","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04356v3","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04356","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_12","alias_value":"ZUB2YOHYY5L3","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZUB2YOHYY5L3E7B7","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZUB2YOHY","created_at":"2026-07-05T01:34:03Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:ZUB2YOHYY5L3E7B7GTBGYPTZW7","target":"record","payload":{"canonical_record":{"source":{"id":"1812.04356","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-12-11T12:35:36Z","cross_cats_sorted":["stat.ML","stat.TH"],"title_canon_sha256":"3f0301542331eb2230c8583e70867514d0ee4ae7bd569fb22fcc2f85e5d2863d","abstract_canon_sha256":"53b3185cfe134b86dd19d46b31c56c80a9f636b6d16f52b3068dbc7ab8af8835"},"schema_version":"1.0"},"canonical_sha256":"cd03ac38f8c757b27c3f34c26c3e79b7d0dc125363991b80f03d8d0720ddecfd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:34:03.815914Z","signature_b64":"iumEdNtEa4rIxDcoH3yO2ghctqAWn0GjaafmORb59JUVqlia3QKW9dEbo/vKKRsGmsbdyNIhyynnKlXkVYi2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cd03ac38f8c757b27c3f34c26c3e79b7d0dc125363991b80f03d8d0720ddecfd","last_reissued_at":"2026-07-05T01:34:03.815513Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:34:03.815513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1812.04356","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:34:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mH2fiSKa7YofVwm5Oywf4S3PtaQq3oGZ27LFw4TNx08/1R7JvNQxQG8BXWo+/zTH+VA+6mKzvg4MvaLipBJvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T08:45:57.636623Z"},"content_sha256":"4f75fc2d09a83b75d1b2d937efff20897838770f0aba51877c76f69dd6c01c37","schema_version":"1.0","event_id":"sha256:4f75fc2d09a83b75d1b2d937efff20897838770f0aba51877c76f69dd6c01c37"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:ZUB2YOHYY5L3E7B7GTBGYPTZW7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Robust Bregman Clustering","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.ML","stat.TH"],"primary_cat":"math.ST","authors_text":"Aur\\'elie Fischer (LPSM (UMR\\_8001)), Claire Br\\'echeteau (ECN, Cl\\'ement Levrard (DATASHAPE, DATASHAPE), LMJL, LPSM (UMR\\_8001))","submitted_at":"2018-12-11T12:35:36Z","abstract_excerpt":"Using a trimming approach, we investigate a k-means type method based on Bregman divergences for clustering data possibly corrupted with clutter noise. The main interest of Bregman divergences is that the standard Lloyd algorithm adapts to these distortion measures, and they are well-suited for clustering data sampled according to mixture models from exponential families. We prove that there exists an optimal codebook, and that an empirically optimal codebook converges a.s. to an optimal codebook in the distortion sense. Moreover, we obtain the sub-Gaussian rate of convergence for k-means 1 $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04356","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1812.04356/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:34:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X7EAkTuPS7fzhfRAoLQRfLxng4NM0RYoxIxrKFQBWrOzLsL67pbFSQmffWnAPLfqkqT4XHGTvSaPQKe/XM/eCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-06T08:45:57.636995Z"},"content_sha256":"92c20b3a967e80437f6ef00581d38db4ab7330015ecd3baf0463c2976e07dbc3","schema_version":"1.0","event_id":"sha256:92c20b3a967e80437f6ef00581d38db4ab7330015ecd3baf0463c2976e07dbc3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/bundle.json","state_url":"https://pith.science/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/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":"2026-07-06T08:45:57Z","links":{"resolver":"https://pith.science/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7","bundle":"https://pith.science/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/bundle.json","state":"https://pith.science/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZUB2YOHYY5L3E7B7GTBGYPTZW7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:ZUB2YOHYY5L3E7B7GTBGYPTZW7","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":"53b3185cfe134b86dd19d46b31c56c80a9f636b6d16f52b3068dbc7ab8af8835","cross_cats_sorted":["stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-12-11T12:35:36Z","title_canon_sha256":"3f0301542331eb2230c8583e70867514d0ee4ae7bd569fb22fcc2f85e5d2863d"},"schema_version":"1.0","source":{"id":"1812.04356","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1812.04356","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"arxiv_version","alias_value":"1812.04356v3","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.04356","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_12","alias_value":"ZUB2YOHYY5L3","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_16","alias_value":"ZUB2YOHYY5L3E7B7","created_at":"2026-07-05T01:34:03Z"},{"alias_kind":"pith_short_8","alias_value":"ZUB2YOHY","created_at":"2026-07-05T01:34:03Z"}],"graph_snapshots":[{"event_id":"sha256:92c20b3a967e80437f6ef00581d38db4ab7330015ecd3baf0463c2976e07dbc3","target":"graph","created_at":"2026-07-05T01:34:03Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1812.04356/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Using a trimming approach, we investigate a k-means type method based on Bregman divergences for clustering data possibly corrupted with clutter noise. The main interest of Bregman divergences is that the standard Lloyd algorithm adapts to these distortion measures, and they are well-suited for clustering data sampled according to mixture models from exponential families. We prove that there exists an optimal codebook, and that an empirically optimal codebook converges a.s. to an optimal codebook in the distortion sense. Moreover, we obtain the sub-Gaussian rate of convergence for k-means 1 $\\","authors_text":"Aur\\'elie Fischer (LPSM (UMR\\_8001)), Claire Br\\'echeteau (ECN, Cl\\'ement Levrard (DATASHAPE, DATASHAPE), LMJL, LPSM (UMR\\_8001))","cross_cats":["stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-12-11T12:35:36Z","title":"Robust Bregman Clustering"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.04356","kind":"arxiv","version":3},"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":"sha256:4f75fc2d09a83b75d1b2d937efff20897838770f0aba51877c76f69dd6c01c37","target":"record","created_at":"2026-07-05T01:34:03Z","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":"53b3185cfe134b86dd19d46b31c56c80a9f636b6d16f52b3068dbc7ab8af8835","cross_cats_sorted":["stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-12-11T12:35:36Z","title_canon_sha256":"3f0301542331eb2230c8583e70867514d0ee4ae7bd569fb22fcc2f85e5d2863d"},"schema_version":"1.0","source":{"id":"1812.04356","kind":"arxiv","version":3}},"canonical_sha256":"cd03ac38f8c757b27c3f34c26c3e79b7d0dc125363991b80f03d8d0720ddecfd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cd03ac38f8c757b27c3f34c26c3e79b7d0dc125363991b80f03d8d0720ddecfd","first_computed_at":"2026-07-05T01:34:03.815513Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:34:03.815513Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iumEdNtEa4rIxDcoH3yO2ghctqAWn0GjaafmORb59JUVqlia3QKW9dEbo/vKKRsGmsbdyNIhyynnKlXkVYi2BQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:34:03.815914Z","signed_message":"canonical_sha256_bytes"},"source_id":"1812.04356","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f75fc2d09a83b75d1b2d937efff20897838770f0aba51877c76f69dd6c01c37","sha256:92c20b3a967e80437f6ef00581d38db4ab7330015ecd3baf0463c2976e07dbc3"],"state_sha256":"be0b3fd1f14a91146a7195c7c463bbeba3e320690c9d9c42c9d59c69c05923b7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zQvut1E0TLZfitTL4CLgAemkBRvQZ9eQFqTWsajFkSeN3LoKXEOklQ39s2DOKv3fz22Kn+y0rHjrU9C3xRQ3Dg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-06T08:45:57.638935Z","bundle_sha256":"cbbf1639a99a9ac994421bce2d35bf6bad37c1a061e4818f547a6c29f32c5012"}}