{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:HJ6K77XRGGY2J55QY574TAD6JE","short_pith_number":"pith:HJ6K77XR","canonical_record":{"source":{"id":"1605.07051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T15:14:09Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"5d2a0a5a8dd3a6fb384d77c4273df2afcfa14a9868c241679f64ed70dea24a91","abstract_canon_sha256":"c8cc02b52425072d71e554420d4a9071151b7947a066f63c15c4da557a965076"},"schema_version":"1.0"},"canonical_sha256":"3a7caffef131b1a4f7b0c77fc9807e4900c22bbd245c5faa63c50882df12cf52","source":{"kind":"arxiv","id":"1605.07051","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.07051","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"arxiv_version","alias_value":"1605.07051v2","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.07051","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"pith_short_12","alias_value":"HJ6K77XRGGY2","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HJ6K77XRGGY2J55Q","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HJ6K77XR","created_at":"2026-05-18T12:30:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:HJ6K77XRGGY2J55QY574TAD6JE","target":"record","payload":{"canonical_record":{"source":{"id":"1605.07051","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T15:14:09Z","cross_cats_sorted":["cs.LG"],"title_canon_sha256":"5d2a0a5a8dd3a6fb384d77c4273df2afcfa14a9868c241679f64ed70dea24a91","abstract_canon_sha256":"c8cc02b52425072d71e554420d4a9071151b7947a066f63c15c4da557a965076"},"schema_version":"1.0"},"canonical_sha256":"3a7caffef131b1a4f7b0c77fc9807e4900c22bbd245c5faa63c50882df12cf52","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:30.897977Z","signature_b64":"nwOH/52cv56XDhoo/zOhE84b6k2mQKzvFdkw7mJqzWeRSpvygzt8vRn9w5ybBmUOybUxKuP97S2mg026XQ18Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a7caffef131b1a4f7b0c77fc9807e4900c22bbd245c5faa63c50882df12cf52","last_reissued_at":"2026-05-18T00:57:30.897558Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:30.897558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1605.07051","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":"2026-05-18T00:57:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"k138hunbLFVEoAojkUwmI9cgBM1JVvednzPRF0vALamP80Eiu1b8BNx/FZrI0RgGCPxfZ1rdRgivPcQ5HJw1CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:06:37.069222Z"},"content_sha256":"5df2d7c399447c44c1ec0486b214cb96b9eec07787b329052939c93a14308cea","schema_version":"1.0","event_id":"sha256:5df2d7c399447c44c1ec0486b214cb96b9eec07787b329052939c93a14308cea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:HJ6K77XRGGY2J55QY574TAD6JE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"stat.ML","authors_text":"John Lafferty, Qinqing Zheng","submitted_at":"2016-05-23T15:14:09Z","abstract_excerpt":"We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( \\mu r^2 \\kappa^2 n \\max(\\mu, \\log n))$ random observations of a $n_1 \\times n_2$ $\\mu$-incoherent matrix of rank $r$ and condition number $\\kappa$, where $n = \\max(n_1, n_2)$, the algorithm linearly converges to the global optimum with high probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07051","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":"2026-05-18T00:57:30Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xYAer4DO0S8Hm1/hiNlWRg4x3uRIizN4CPByy6Lair+GvQiVP8bnzeBidVBF6dscjAEOtuX4ptycOzTM4Pt9BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T16:06:37.069590Z"},"content_sha256":"01406b0e87f9dcb38c0fdbae500597978358ce6c6c3831c375543753c4d9dd99","schema_version":"1.0","event_id":"sha256:01406b0e87f9dcb38c0fdbae500597978358ce6c6c3831c375543753c4d9dd99"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HJ6K77XRGGY2J55QY574TAD6JE/bundle.json","state_url":"https://pith.science/pith/HJ6K77XRGGY2J55QY574TAD6JE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HJ6K77XRGGY2J55QY574TAD6JE/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":"2026-05-28T16:06:37Z","links":{"resolver":"https://pith.science/pith/HJ6K77XRGGY2J55QY574TAD6JE","bundle":"https://pith.science/pith/HJ6K77XRGGY2J55QY574TAD6JE/bundle.json","state":"https://pith.science/pith/HJ6K77XRGGY2J55QY574TAD6JE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HJ6K77XRGGY2J55QY574TAD6JE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:HJ6K77XRGGY2J55QY574TAD6JE","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":"c8cc02b52425072d71e554420d4a9071151b7947a066f63c15c4da557a965076","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T15:14:09Z","title_canon_sha256":"5d2a0a5a8dd3a6fb384d77c4273df2afcfa14a9868c241679f64ed70dea24a91"},"schema_version":"1.0","source":{"id":"1605.07051","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1605.07051","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"arxiv_version","alias_value":"1605.07051v2","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1605.07051","created_at":"2026-05-18T00:57:30Z"},{"alias_kind":"pith_short_12","alias_value":"HJ6K77XRGGY2","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_16","alias_value":"HJ6K77XRGGY2J55Q","created_at":"2026-05-18T12:30:19Z"},{"alias_kind":"pith_short_8","alias_value":"HJ6K77XR","created_at":"2026-05-18T12:30:19Z"}],"graph_snapshots":[{"event_id":"sha256:01406b0e87f9dcb38c0fdbae500597978358ce6c6c3831c375543753c4d9dd99","target":"graph","created_at":"2026-05-18T00:57:30Z","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 address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( \\mu r^2 \\kappa^2 n \\max(\\mu, \\log n))$ random observations of a $n_1 \\times n_2$ $\\mu$-incoherent matrix of rank $r$ and condition number $\\kappa$, where $n = \\max(n_1, n_2)$, the algorithm linearly converges to the global optimum with high probability.","authors_text":"John Lafferty, Qinqing Zheng","cross_cats":["cs.LG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T15:14:09Z","title":"Convergence Analysis for Rectangular Matrix Completion Using Burer-Monteiro Factorization and Gradient Descent"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07051","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":"sha256:5df2d7c399447c44c1ec0486b214cb96b9eec07787b329052939c93a14308cea","target":"record","created_at":"2026-05-18T00:57:30Z","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":"c8cc02b52425072d71e554420d4a9071151b7947a066f63c15c4da557a965076","cross_cats_sorted":["cs.LG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-05-23T15:14:09Z","title_canon_sha256":"5d2a0a5a8dd3a6fb384d77c4273df2afcfa14a9868c241679f64ed70dea24a91"},"schema_version":"1.0","source":{"id":"1605.07051","kind":"arxiv","version":2}},"canonical_sha256":"3a7caffef131b1a4f7b0c77fc9807e4900c22bbd245c5faa63c50882df12cf52","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3a7caffef131b1a4f7b0c77fc9807e4900c22bbd245c5faa63c50882df12cf52","first_computed_at":"2026-05-18T00:57:30.897558Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:30.897558Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nwOH/52cv56XDhoo/zOhE84b6k2mQKzvFdkw7mJqzWeRSpvygzt8vRn9w5ybBmUOybUxKuP97S2mg026XQ18Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:30.897977Z","signed_message":"canonical_sha256_bytes"},"source_id":"1605.07051","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5df2d7c399447c44c1ec0486b214cb96b9eec07787b329052939c93a14308cea","sha256:01406b0e87f9dcb38c0fdbae500597978358ce6c6c3831c375543753c4d9dd99"],"state_sha256":"2454da335f23fd60b514f82631542becbb4c2e9f9144465b9c2a164e840d4ae3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FteBGxUwk12RXKMBVgVN+fZKkbOZcYHHOGGobJCiBM17SyVlwHfrHRqH8pwIIt8qx+VzRyqcFkqWzw2Ub1ZvAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T16:06:37.071741Z","bundle_sha256":"78aa6826d7a374b62f3f522c175904c484bb7b220beb7dcd978fd6b87fe4b483"}}