{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7VKQRQQ777ZQIZXJTISGT325ZR","short_pith_number":"pith:7VKQRQQ7","canonical_record":{"source":{"id":"1010.1763","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2010-10-08T18:53:27Z","cross_cats_sorted":[],"title_canon_sha256":"1d25f08ad2e9e94e1794e89c353214024606dd7c19625d4d78c1f13980b2457d","abstract_canon_sha256":"64ecf7c99a53ee6d53c14bf959e62b7840dadc08d9eec28282c47028f6217389"},"schema_version":"1.0"},"canonical_sha256":"fd5508c21ffff30466e99a2469ef5dcc4afe4bee2e8bbb1a7e096c2c42c0a6da","source":{"kind":"arxiv","id":"1010.1763","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.1763","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"arxiv_version","alias_value":"1010.1763v3","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1763","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"pith_short_12","alias_value":"7VKQRQQ777ZQ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7VKQRQQ777ZQIZXJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7VKQRQQ7","created_at":"2026-05-18T12:26:05Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7VKQRQQ777ZQIZXJTISGT325ZR","target":"record","payload":{"canonical_record":{"source":{"id":"1010.1763","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2010-10-08T18:53:27Z","cross_cats_sorted":[],"title_canon_sha256":"1d25f08ad2e9e94e1794e89c353214024606dd7c19625d4d78c1f13980b2457d","abstract_canon_sha256":"64ecf7c99a53ee6d53c14bf959e62b7840dadc08d9eec28282c47028f6217389"},"schema_version":"1.0"},"canonical_sha256":"fd5508c21ffff30466e99a2469ef5dcc4afe4bee2e8bbb1a7e096c2c42c0a6da","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:16.183672Z","signature_b64":"KD6DzUXciCysNwZmIDDQ/joXy0Fn24WXjUnFQCUSq4eCjfAtutEuWrpHi8OI9vN4o6bUS6xYG04V0xN1CgjCAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd5508c21ffff30466e99a2469ef5dcc4afe4bee2e8bbb1a7e096c2c42c0a6da","last_reissued_at":"2026-05-18T04:27:16.182980Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:16.182980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1010.1763","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":"2026-05-18T04:27:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yyOpHxyn0QFYlAnj5NTJ3xm3FKWn61EW/nsNS1g/b/UoZvkpq8NafvL1RYkBsAxxCj/xuDbimQBBkU1s4H8qBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T20:29:04.070448Z"},"content_sha256":"506fe2fb61fed64dc7147269c50d615e51086a3945271547228dbb21cc42ba62","schema_version":"1.0","event_id":"sha256:506fe2fb61fed64dc7147269c50d615e51086a3945271547228dbb21cc42ba62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7VKQRQQ777ZQIZXJTISGT325ZR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Algorithms for nonnegative matrix factorization with the beta-divergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"C\\'edric F\\'evotte (LTCI), J\\'er\\^ome Idier (IRCCyN)","submitted_at":"2010-10-08T18:53:27Z","abstract_excerpt":"This paper describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parametrized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (beta = 2,1,0, respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization (MM) algorithm that leads to multiplicative updates, which differ from standard h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1763","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:27:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FcVB67rGqtCCvmgEGEkCy7gRvrbKZorwTr6Ecz/gZub4MsvHu0a/srsO92DXihK0SXnmpFWwrnQA/0Rxoox2Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T20:29:04.070801Z"},"content_sha256":"e5cd6af2cc11e9dca97f5632901c72a4f2540007793031a27ef2bc9e08dcebb6","schema_version":"1.0","event_id":"sha256:e5cd6af2cc11e9dca97f5632901c72a4f2540007793031a27ef2bc9e08dcebb6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7VKQRQQ777ZQIZXJTISGT325ZR/bundle.json","state_url":"https://pith.science/pith/7VKQRQQ777ZQIZXJTISGT325ZR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7VKQRQQ777ZQIZXJTISGT325ZR/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":"2026-06-30T20:29:04Z","links":{"resolver":"https://pith.science/pith/7VKQRQQ777ZQIZXJTISGT325ZR","bundle":"https://pith.science/pith/7VKQRQQ777ZQIZXJTISGT325ZR/bundle.json","state":"https://pith.science/pith/7VKQRQQ777ZQIZXJTISGT325ZR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7VKQRQQ777ZQIZXJTISGT325ZR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7VKQRQQ777ZQIZXJTISGT325ZR","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":"64ecf7c99a53ee6d53c14bf959e62b7840dadc08d9eec28282c47028f6217389","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2010-10-08T18:53:27Z","title_canon_sha256":"1d25f08ad2e9e94e1794e89c353214024606dd7c19625d4d78c1f13980b2457d"},"schema_version":"1.0","source":{"id":"1010.1763","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1010.1763","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"arxiv_version","alias_value":"1010.1763v3","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1763","created_at":"2026-05-18T04:27:16Z"},{"alias_kind":"pith_short_12","alias_value":"7VKQRQQ777ZQ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_16","alias_value":"7VKQRQQ777ZQIZXJ","created_at":"2026-05-18T12:26:05Z"},{"alias_kind":"pith_short_8","alias_value":"7VKQRQQ7","created_at":"2026-05-18T12:26:05Z"}],"graph_snapshots":[{"event_id":"sha256:e5cd6af2cc11e9dca97f5632901c72a4f2540007793031a27ef2bc9e08dcebb6","target":"graph","created_at":"2026-05-18T04:27:16Z","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":"This paper describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parametrized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (beta = 2,1,0, respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization (MM) algorithm that leads to multiplicative updates, which differ from standard h","authors_text":"C\\'edric F\\'evotte (LTCI), J\\'er\\^ome Idier (IRCCyN)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2010-10-08T18:53:27Z","title":"Algorithms for nonnegative matrix factorization with the beta-divergence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1763","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":"sha256:506fe2fb61fed64dc7147269c50d615e51086a3945271547228dbb21cc42ba62","target":"record","created_at":"2026-05-18T04:27:16Z","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":"64ecf7c99a53ee6d53c14bf959e62b7840dadc08d9eec28282c47028f6217389","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2010-10-08T18:53:27Z","title_canon_sha256":"1d25f08ad2e9e94e1794e89c353214024606dd7c19625d4d78c1f13980b2457d"},"schema_version":"1.0","source":{"id":"1010.1763","kind":"arxiv","version":3}},"canonical_sha256":"fd5508c21ffff30466e99a2469ef5dcc4afe4bee2e8bbb1a7e096c2c42c0a6da","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd5508c21ffff30466e99a2469ef5dcc4afe4bee2e8bbb1a7e096c2c42c0a6da","first_computed_at":"2026-05-18T04:27:16.182980Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:16.182980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KD6DzUXciCysNwZmIDDQ/joXy0Fn24WXjUnFQCUSq4eCjfAtutEuWrpHi8OI9vN4o6bUS6xYG04V0xN1CgjCAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:16.183672Z","signed_message":"canonical_sha256_bytes"},"source_id":"1010.1763","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:506fe2fb61fed64dc7147269c50d615e51086a3945271547228dbb21cc42ba62","sha256:e5cd6af2cc11e9dca97f5632901c72a4f2540007793031a27ef2bc9e08dcebb6"],"state_sha256":"c0a63855732895d2613e5f6b89811f51bb90f3b2bfd1147d0fb28d4fe5bd2511"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r3KTBoyF/vCCl/f1EBDjo7mfk/EmN0WA/22Y7C4fZ4FVqxgQjsKohg/ja0FgvSoiyZLiq0BltM2ePq5DUk8LDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T20:29:04.072767Z","bundle_sha256":"6a2890fcdd07a868d369d2712c8577dd57d28baf07e8ae27b039de0c1bc51ac7"}}