{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:MKYJE26BVB3SREVQUYQBNBCJVH","short_pith_number":"pith:MKYJE26B","canonical_record":{"source":{"id":"1608.00075","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-07-30T06:07:38Z","cross_cats_sorted":["cs.IT","cs.NA","math.IT","math.OC"],"title_canon_sha256":"1ac3a3e8c2e281b76c01637f8b637110437a6403db64f910170ab321fa1f50ef","abstract_canon_sha256":"5f86a081636ccb53282ccf01ca1824235aa2bce08c065002b15b559776108bf8"},"schema_version":"1.0"},"canonical_sha256":"62b0926bc1a8772892b0a620168449a9d39a95d3fc05adb751b88280f0f4a8f4","source":{"kind":"arxiv","id":"1608.00075","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.00075","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"arxiv_version","alias_value":"1608.00075v2","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.00075","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"pith_short_12","alias_value":"MKYJE26BVB3S","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MKYJE26BVB3SREVQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MKYJE26B","created_at":"2026-05-18T12:30:32Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:MKYJE26BVB3SREVQUYQBNBCJVH","target":"record","payload":{"canonical_record":{"source":{"id":"1608.00075","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-07-30T06:07:38Z","cross_cats_sorted":["cs.IT","cs.NA","math.IT","math.OC"],"title_canon_sha256":"1ac3a3e8c2e281b76c01637f8b637110437a6403db64f910170ab321fa1f50ef","abstract_canon_sha256":"5f86a081636ccb53282ccf01ca1824235aa2bce08c065002b15b559776108bf8"},"schema_version":"1.0"},"canonical_sha256":"62b0926bc1a8772892b0a620168449a9d39a95d3fc05adb751b88280f0f4a8f4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:37.902204Z","signature_b64":"FpQKo6FF57lh28y+Muah9S211GzCrZK+pOXf/6RcbMDZfZrRg2LzQOVtgbygHRDYPhhxxtrZWMkaPSWCgLl5CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62b0926bc1a8772892b0a620168449a9d39a95d3fc05adb751b88280f0f4a8f4","last_reissued_at":"2026-05-18T01:08:37.901590Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:37.901590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.00075","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:08:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vKfEjqPR0b16u+B/w6jgpBi2599Ml6U+YSNlCYliHLigkM8D9lpaeSfsu+54xG4ebHIMUBwjlugcHFjEIUWlDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:14:36.348229Z"},"content_sha256":"e6562083d93c16b8784ee95daff06318ee3f84ae4eb4a05ca499d4141a4b5c0b","schema_version":"1.0","event_id":"sha256:e6562083d93c16b8784ee95daff06318ee3f84ae4eb4a05ca499d4141a4b5c0b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:MKYJE26BVB3SREVQUYQBNBCJVH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Online Nonnegative Matrix Factorization with General Divergences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.NA","math.IT","math.OC"],"primary_cat":"stat.ML","authors_text":"Huan Xu, Renbo Zhao, Vincent Y. F. Tan","submitted_at":"2016-07-30T06:07:38Z","abstract_excerpt":"We develop a unified and systematic framework for performing online nonnegative matrix factorization under a wide variety of important divergences. The online nature of our algorithm makes it particularly amenable to large-scale data. We prove that the sequence of learned dictionaries converges almost surely to the set of critical points of the expected loss function. We do so by leveraging the theory of stochastic approximations and projected dynamical systems. This result substantially generalizes the previous results obtained only for the squared-$\\ell_2$ loss. Moreover, the novel technique"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00075","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":""},"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":"2026-05-18T01:08:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9hGpN/gf3IYybdA5Yn34zkyCvA9orDZ+L9X7roG32Q5Z33WOVNDnXUmHmbxpdz8SScyP5b7HGVtJZW/OupKuBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T12:14:36.348591Z"},"content_sha256":"25e13e9ffe996a4c4f1e74b43f6869839dc4ba9e180f702b7e166140c3ecdfc0","schema_version":"1.0","event_id":"sha256:25e13e9ffe996a4c4f1e74b43f6869839dc4ba9e180f702b7e166140c3ecdfc0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MKYJE26BVB3SREVQUYQBNBCJVH/bundle.json","state_url":"https://pith.science/pith/MKYJE26BVB3SREVQUYQBNBCJVH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MKYJE26BVB3SREVQUYQBNBCJVH/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":"2026-06-01T12:14:36Z","links":{"resolver":"https://pith.science/pith/MKYJE26BVB3SREVQUYQBNBCJVH","bundle":"https://pith.science/pith/MKYJE26BVB3SREVQUYQBNBCJVH/bundle.json","state":"https://pith.science/pith/MKYJE26BVB3SREVQUYQBNBCJVH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MKYJE26BVB3SREVQUYQBNBCJVH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:MKYJE26BVB3SREVQUYQBNBCJVH","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":"5f86a081636ccb53282ccf01ca1824235aa2bce08c065002b15b559776108bf8","cross_cats_sorted":["cs.IT","cs.NA","math.IT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-07-30T06:07:38Z","title_canon_sha256":"1ac3a3e8c2e281b76c01637f8b637110437a6403db64f910170ab321fa1f50ef"},"schema_version":"1.0","source":{"id":"1608.00075","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.00075","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"arxiv_version","alias_value":"1608.00075v2","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.00075","created_at":"2026-05-18T01:08:37Z"},{"alias_kind":"pith_short_12","alias_value":"MKYJE26BVB3S","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_16","alias_value":"MKYJE26BVB3SREVQ","created_at":"2026-05-18T12:30:32Z"},{"alias_kind":"pith_short_8","alias_value":"MKYJE26B","created_at":"2026-05-18T12:30:32Z"}],"graph_snapshots":[{"event_id":"sha256:25e13e9ffe996a4c4f1e74b43f6869839dc4ba9e180f702b7e166140c3ecdfc0","target":"graph","created_at":"2026-05-18T01:08:37Z","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":"We develop a unified and systematic framework for performing online nonnegative matrix factorization under a wide variety of important divergences. The online nature of our algorithm makes it particularly amenable to large-scale data. We prove that the sequence of learned dictionaries converges almost surely to the set of critical points of the expected loss function. We do so by leveraging the theory of stochastic approximations and projected dynamical systems. This result substantially generalizes the previous results obtained only for the squared-$\\ell_2$ loss. Moreover, the novel technique","authors_text":"Huan Xu, Renbo Zhao, Vincent Y. F. Tan","cross_cats":["cs.IT","cs.NA","math.IT","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-07-30T06:07:38Z","title":"Online Nonnegative Matrix Factorization with General Divergences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00075","kind":"arxiv","version":2},"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:e6562083d93c16b8784ee95daff06318ee3f84ae4eb4a05ca499d4141a4b5c0b","target":"record","created_at":"2026-05-18T01:08:37Z","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":"5f86a081636ccb53282ccf01ca1824235aa2bce08c065002b15b559776108bf8","cross_cats_sorted":["cs.IT","cs.NA","math.IT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-07-30T06:07:38Z","title_canon_sha256":"1ac3a3e8c2e281b76c01637f8b637110437a6403db64f910170ab321fa1f50ef"},"schema_version":"1.0","source":{"id":"1608.00075","kind":"arxiv","version":2}},"canonical_sha256":"62b0926bc1a8772892b0a620168449a9d39a95d3fc05adb751b88280f0f4a8f4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62b0926bc1a8772892b0a620168449a9d39a95d3fc05adb751b88280f0f4a8f4","first_computed_at":"2026-05-18T01:08:37.901590Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:08:37.901590Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FpQKo6FF57lh28y+Muah9S211GzCrZK+pOXf/6RcbMDZfZrRg2LzQOVtgbygHRDYPhhxxtrZWMkaPSWCgLl5CA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:08:37.902204Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.00075","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6562083d93c16b8784ee95daff06318ee3f84ae4eb4a05ca499d4141a4b5c0b","sha256:25e13e9ffe996a4c4f1e74b43f6869839dc4ba9e180f702b7e166140c3ecdfc0"],"state_sha256":"c5fe425bab488faac6e326ae4c3c7e6fcdc54e30823d8fdbe0ec5e9e04ae6d41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QrmkNUf4pB2ZIx142z5Nhu3HZ/c4GKWre9xsz7rQWdhvXhlC2IuR5KdgeaSLuA1EOmxAoOU8jB6okrQy+gJXBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T12:14:36.350758Z","bundle_sha256":"c0faf5a18573c9f87de564f63a78aec68cb3aaa7caf51664ae7ff5bf5afd82b3"}}