{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:RRR6Z2XPEZFP7GNTYHQIBVNQXB","short_pith_number":"pith:RRR6Z2XP","canonical_record":{"source":{"id":"1606.01316","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-06-04T02:12:13Z","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"title_canon_sha256":"7023059f02a421e4bec3784b2d366e1a9d57ae76e1bdd404ff5e9b74c18f614e","abstract_canon_sha256":"14436074d6859f13cb878b64cf264cb8c3014e143e594ce3aa777ac0da5ac426"},"schema_version":"1.0"},"canonical_sha256":"8c63eceaef264aff99b3c1e080d5b0b84a78260af95923a8eb55432e8be47f66","source":{"kind":"arxiv","id":"1606.01316","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.01316","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"arxiv_version","alias_value":"1606.01316v3","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.01316","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"pith_short_12","alias_value":"RRR6Z2XPEZFP","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RRR6Z2XPEZFP7GNT","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RRR6Z2XP","created_at":"2026-05-18T12:30:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:RRR6Z2XPEZFP7GNTYHQIBVNQXB","target":"record","payload":{"canonical_record":{"source":{"id":"1606.01316","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-06-04T02:12:13Z","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"title_canon_sha256":"7023059f02a421e4bec3784b2d366e1a9d57ae76e1bdd404ff5e9b74c18f614e","abstract_canon_sha256":"14436074d6859f13cb878b64cf264cb8c3014e143e594ce3aa777ac0da5ac426"},"schema_version":"1.0"},"canonical_sha256":"8c63eceaef264aff99b3c1e080d5b0b84a78260af95923a8eb55432e8be47f66","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:32.491202Z","signature_b64":"62dXP9PGX9DhjLn6KqxfNAj3BDe8Sah6ufWemqH4GNK7J6UKkbScVa6Mc3LCipIbCyFuA/HHxpn1k+2XJbn2AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c63eceaef264aff99b3c1e080d5b0b84a78260af95923a8eb55432e8be47f66","last_reissued_at":"2026-05-18T01:03:32.490589Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:32.490589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.01316","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":"2026-05-18T01:03:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7cTGuI3k6rET2PTWu0cx4l542NhpQZr/n8o5eDr79FTbs5NoHf8bzDXDEoUn18QWlYfB7kMYA5FJWkMcdlUuDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:27:37.631100Z"},"content_sha256":"b685fa60a084b077b47a5939c9cd26087107b7b341aa33f26bbd359559c276f6","schema_version":"1.0","event_id":"sha256:b685fa60a084b077b47a5939c9cd26087107b7b341aa33f26bbd359559c276f6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:RRR6Z2XPEZFP7GNTYHQIBVNQXB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"primary_cat":"stat.ML","authors_text":"Anastasios Kyrillidis, Constantine Caramanis, Dohyung Park, Srinadh Bhojanapalli, Sujay Sanghavi","submitted_at":"2016-06-04T02:12:13Z","abstract_excerpt":"We study the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-definite (PSD) constraints and specific matrix norm-constraints. Such criteria appear in quantum state tomography and phase retrieval applications.\n  We show that non-convex projected gradient descent favors local linear convergence in the factored space. We build our theory on a nov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01316","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":"2026-05-18T01:03:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Txm5TqkMcLLE7OjHFjPi5BU7vkJqY1q1Tl8ZnyTkxJAPyXV63OcPR9jeeEnmXXOC5F9phqW2AErEfdLMTdDrAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T15:27:37.631771Z"},"content_sha256":"2fc019c914cf79bc84eed42d3aaadc3e17504c919add4a9dd7545ef50aebd107","schema_version":"1.0","event_id":"sha256:2fc019c914cf79bc84eed42d3aaadc3e17504c919add4a9dd7545ef50aebd107"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/bundle.json","state_url":"https://pith.science/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/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":"2026-05-25T15:27:37Z","links":{"resolver":"https://pith.science/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB","bundle":"https://pith.science/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/bundle.json","state":"https://pith.science/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RRR6Z2XPEZFP7GNTYHQIBVNQXB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:RRR6Z2XPEZFP7GNTYHQIBVNQXB","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":"14436074d6859f13cb878b64cf264cb8c3014e143e594ce3aa777ac0da5ac426","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-06-04T02:12:13Z","title_canon_sha256":"7023059f02a421e4bec3784b2d366e1a9d57ae76e1bdd404ff5e9b74c18f614e"},"schema_version":"1.0","source":{"id":"1606.01316","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.01316","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"arxiv_version","alias_value":"1606.01316v3","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.01316","created_at":"2026-05-18T01:03:32Z"},{"alias_kind":"pith_short_12","alias_value":"RRR6Z2XPEZFP","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_16","alias_value":"RRR6Z2XPEZFP7GNT","created_at":"2026-05-18T12:30:41Z"},{"alias_kind":"pith_short_8","alias_value":"RRR6Z2XP","created_at":"2026-05-18T12:30:41Z"}],"graph_snapshots":[{"event_id":"sha256:2fc019c914cf79bc84eed42d3aaadc3e17504c919add4a9dd7545ef50aebd107","target":"graph","created_at":"2026-05-18T01:03:32Z","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 the projected gradient descent method on low-rank matrix problems with a strongly convex objective. We use the Burer-Monteiro factorization approach to implicitly enforce low-rankness; such factorization introduces non-convexity in the objective. We focus on constraint sets that include both positive semi-definite (PSD) constraints and specific matrix norm-constraints. Such criteria appear in quantum state tomography and phase retrieval applications.\n  We show that non-convex projected gradient descent favors local linear convergence in the factored space. We build our theory on a nov","authors_text":"Anastasios Kyrillidis, Constantine Caramanis, Dohyung Park, Srinadh Bhojanapalli, Sujay Sanghavi","cross_cats":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-06-04T02:12:13Z","title":"Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.01316","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:b685fa60a084b077b47a5939c9cd26087107b7b341aa33f26bbd359559c276f6","target":"record","created_at":"2026-05-18T01:03:32Z","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":"14436074d6859f13cb878b64cf264cb8c3014e143e594ce3aa777ac0da5ac426","cross_cats_sorted":["cs.DS","cs.IT","cs.NA","math.IT","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"stat.ML","submitted_at":"2016-06-04T02:12:13Z","title_canon_sha256":"7023059f02a421e4bec3784b2d366e1a9d57ae76e1bdd404ff5e9b74c18f614e"},"schema_version":"1.0","source":{"id":"1606.01316","kind":"arxiv","version":3}},"canonical_sha256":"8c63eceaef264aff99b3c1e080d5b0b84a78260af95923a8eb55432e8be47f66","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c63eceaef264aff99b3c1e080d5b0b84a78260af95923a8eb55432e8be47f66","first_computed_at":"2026-05-18T01:03:32.490589Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:03:32.490589Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"62dXP9PGX9DhjLn6KqxfNAj3BDe8Sah6ufWemqH4GNK7J6UKkbScVa6Mc3LCipIbCyFuA/HHxpn1k+2XJbn2AA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:03:32.491202Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.01316","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b685fa60a084b077b47a5939c9cd26087107b7b341aa33f26bbd359559c276f6","sha256:2fc019c914cf79bc84eed42d3aaadc3e17504c919add4a9dd7545ef50aebd107"],"state_sha256":"ef5ab815a60c2c30548ca8a288b06bbed5cafb4a0f40fd4e938d50ca4289c305"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9T07fPb1dsWje2Xp2dktyc9LbfItau4dNVX1jVJwCmFldbJvhHK3GUccUhvZqPH27DcfazrOGz6spgGmCxKcDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T15:27:37.635816Z","bundle_sha256":"ac4747e3717d2448fd388c18a4a0652fec3b6f6aedafe74f8739cfb4d211d607"}}