{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:YOR3PO4TYCAXE3RN3OXR4ZQU44","short_pith_number":"pith:YOR3PO4T","canonical_record":{"source":{"id":"1607.07837","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-26T18:46:21Z","cross_cats_sorted":["cs.DS","cs.LG","math.NA","stat.ML"],"title_canon_sha256":"9add3ad4a6fc2abcba23068aa13568733ac336cf62121657f28343c2d6cc3b93","abstract_canon_sha256":"c82070cecffef215d39ea55988c2a9857a1fecb084f5c89200afd90d43d16fd3"},"schema_version":"1.0"},"canonical_sha256":"c3a3b7bb93c081726e2ddbaf1e6614e72696de08e82329522a5402d67c08c884","source":{"kind":"arxiv","id":"1607.07837","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07837","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07837v4","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07837","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"pith_short_12","alias_value":"YOR3PO4TYCAX","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YOR3PO4TYCAXE3RN","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YOR3PO4T","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:YOR3PO4TYCAXE3RN3OXR4ZQU44","target":"record","payload":{"canonical_record":{"source":{"id":"1607.07837","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-26T18:46:21Z","cross_cats_sorted":["cs.DS","cs.LG","math.NA","stat.ML"],"title_canon_sha256":"9add3ad4a6fc2abcba23068aa13568733ac336cf62121657f28343c2d6cc3b93","abstract_canon_sha256":"c82070cecffef215d39ea55988c2a9857a1fecb084f5c89200afd90d43d16fd3"},"schema_version":"1.0"},"canonical_sha256":"c3a3b7bb93c081726e2ddbaf1e6614e72696de08e82329522a5402d67c08c884","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:20.860376Z","signature_b64":"4+MmY1PKpgwm2ftPfQJ3PS7Z2crNIsP66d8o+ayj/rNuYQirJx77MSrmPUTKiU9YCvbffC7eJHSM1pZzng6hCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c3a3b7bb93c081726e2ddbaf1e6614e72696de08e82329522a5402d67c08c884","last_reissued_at":"2026-05-18T00:46:20.860022Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:20.860022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1607.07837","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:46:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4qllUAhf3hAPFLmyoBKc2exCovUf5FKiTNumvCzmV9MqwqnnL6ANYQoqNLZ3tB0N6fLT7r0DPj1ZhWxz+hQlBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T00:27:26.768581Z"},"content_sha256":"95ddb1050252b84905346155624d7eb4d2e08d6bd57db20b1e496be8c011c083","schema_version":"1.0","event_id":"sha256:95ddb1050252b84905346155624d7eb4d2e08d6bd57db20b1e496be8c011c083"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:YOR3PO4TYCAXE3RN3OXR4ZQU44","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.LG","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Yuanzhi Li, Zeyuan Allen-Zhu","submitted_at":"2016-07-26T18:46:21Z","abstract_excerpt":"We study streaming principal component analysis (PCA), that is to find, in $O(dk)$ space, the top $k$ eigenvectors of a $d\\times d$ hidden matrix $\\bf \\Sigma$ with online vectors drawn from covariance matrix $\\bf \\Sigma$.\n  We provide $\\textit{global}$ convergence for Oja's algorithm which is popularly used in practice but lacks theoretical understanding for $k>1$. We also provide a modified variant $\\mathsf{Oja}^{++}$ that runs $\\textit{even faster}$ than Oja's. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank $k$, and on dimension "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07837","kind":"arxiv","version":4},"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":"2026-05-18T00:46:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"giXjv7DK2KB6PAMCoOyaVQghCNPJEXaWuEO4hbmilFIyeEARBlNGJMAxwAZ3zOA3eT8sDpsOZ8+K/n7zg24KCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-19T00:27:26.769100Z"},"content_sha256":"93d644bc4b1dbf0041ac7322c0f267a21ae05c113e194c4407efb4705476ca7b","schema_version":"1.0","event_id":"sha256:93d644bc4b1dbf0041ac7322c0f267a21ae05c113e194c4407efb4705476ca7b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/bundle.json","state_url":"https://pith.science/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/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":"2026-08-19T00:27:26Z","links":{"resolver":"https://pith.science/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44","bundle":"https://pith.science/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/bundle.json","state":"https://pith.science/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YOR3PO4TYCAXE3RN3OXR4ZQU44/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:YOR3PO4TYCAXE3RN3OXR4ZQU44","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":"c82070cecffef215d39ea55988c2a9857a1fecb084f5c89200afd90d43d16fd3","cross_cats_sorted":["cs.DS","cs.LG","math.NA","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-26T18:46:21Z","title_canon_sha256":"9add3ad4a6fc2abcba23068aa13568733ac336cf62121657f28343c2d6cc3b93"},"schema_version":"1.0","source":{"id":"1607.07837","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1607.07837","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"arxiv_version","alias_value":"1607.07837v4","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1607.07837","created_at":"2026-05-18T00:46:20Z"},{"alias_kind":"pith_short_12","alias_value":"YOR3PO4TYCAX","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"YOR3PO4TYCAXE3RN","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"YOR3PO4T","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:93d644bc4b1dbf0041ac7322c0f267a21ae05c113e194c4407efb4705476ca7b","target":"graph","created_at":"2026-05-18T00:46:20Z","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 study streaming principal component analysis (PCA), that is to find, in $O(dk)$ space, the top $k$ eigenvectors of a $d\\times d$ hidden matrix $\\bf \\Sigma$ with online vectors drawn from covariance matrix $\\bf \\Sigma$.\n  We provide $\\textit{global}$ convergence for Oja's algorithm which is popularly used in practice but lacks theoretical understanding for $k>1$. We also provide a modified variant $\\mathsf{Oja}^{++}$ that runs $\\textit{even faster}$ than Oja's. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank $k$, and on dimension ","authors_text":"Yuanzhi Li, Zeyuan Allen-Zhu","cross_cats":["cs.DS","cs.LG","math.NA","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-26T18:46:21Z","title":"First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07837","kind":"arxiv","version":4},"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":"sha256:95ddb1050252b84905346155624d7eb4d2e08d6bd57db20b1e496be8c011c083","target":"record","created_at":"2026-05-18T00:46:20Z","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":"c82070cecffef215d39ea55988c2a9857a1fecb084f5c89200afd90d43d16fd3","cross_cats_sorted":["cs.DS","cs.LG","math.NA","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-07-26T18:46:21Z","title_canon_sha256":"9add3ad4a6fc2abcba23068aa13568733ac336cf62121657f28343c2d6cc3b93"},"schema_version":"1.0","source":{"id":"1607.07837","kind":"arxiv","version":4}},"canonical_sha256":"c3a3b7bb93c081726e2ddbaf1e6614e72696de08e82329522a5402d67c08c884","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c3a3b7bb93c081726e2ddbaf1e6614e72696de08e82329522a5402d67c08c884","first_computed_at":"2026-05-18T00:46:20.860022Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:46:20.860022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4+MmY1PKpgwm2ftPfQJ3PS7Z2crNIsP66d8o+ayj/rNuYQirJx77MSrmPUTKiU9YCvbffC7eJHSM1pZzng6hCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:46:20.860376Z","signed_message":"canonical_sha256_bytes"},"source_id":"1607.07837","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:95ddb1050252b84905346155624d7eb4d2e08d6bd57db20b1e496be8c011c083","sha256:93d644bc4b1dbf0041ac7322c0f267a21ae05c113e194c4407efb4705476ca7b"],"state_sha256":"adfc89dce6ad4f06a2120407a06d310bf1f1fec9b0c549bb671f824e33c54c2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZQIjxGYbP69ATw4dPksbtv6gUO1/WcA2acTnt6OqgdYXAScTr3jKpNXSUoZpa14y6AwKNW1zZ8Tvz8rr/D0fDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-19T00:27:26.774726Z","bundle_sha256":"8b5bb4bd9a9b222098ceb51d520381e4526250193df492ccfdc0483a933f33b2"}}