{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:RL3OY33SEJBYHNKVFRTYI45QOH","short_pith_number":"pith:RL3OY33S","canonical_record":{"source":{"id":"1211.1550","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-11-07T13:49:26Z","cross_cats_sorted":["cs.NA","math.OC"],"title_canon_sha256":"f98bf077e32ff5907398da06f7e0d4323be1b4ae6d6bc6dfd8ee097186966c29","abstract_canon_sha256":"b37d4bc35f3ed799bc6d7c1f6ee2afe0cf2337970ec828bb27e5daa8846b495f"},"schema_version":"1.0"},"canonical_sha256":"8af6ec6f72224383b5552c678473b071df987937bbe667d9cec0e51a15dee585","source":{"kind":"arxiv","id":"1211.1550","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1550","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1550v2","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1550","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"pith_short_12","alias_value":"RL3OY33SEJBY","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RL3OY33SEJBYHNKV","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RL3OY33S","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:RL3OY33SEJBYHNKVFRTYI45QOH","target":"record","payload":{"canonical_record":{"source":{"id":"1211.1550","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-11-07T13:49:26Z","cross_cats_sorted":["cs.NA","math.OC"],"title_canon_sha256":"f98bf077e32ff5907398da06f7e0d4323be1b4ae6d6bc6dfd8ee097186966c29","abstract_canon_sha256":"b37d4bc35f3ed799bc6d7c1f6ee2afe0cf2337970ec828bb27e5daa8846b495f"},"schema_version":"1.0"},"canonical_sha256":"8af6ec6f72224383b5552c678473b071df987937bbe667d9cec0e51a15dee585","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:41:02.278047Z","signature_b64":"0enCQdV6FMP9BL/NQTP6qLR584BbKhn/FJcywAhDr30bJEl5cAsZ1lMlOJSQKQqvwx3bFZbOkJp7zdv+dwHWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8af6ec6f72224383b5552c678473b071df987937bbe667d9cec0e51a15dee585","last_reissued_at":"2026-05-18T03:41:02.277438Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:41:02.277438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.1550","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":"2026-05-18T03:41:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gwquEvzM1lZCOCgpTUv7GQj+qROjHSBDBm4jHA0ae4z0hTkt6xPD3lH4+vTAwpywd7f4Fo8IKAQaPAnlkd8nDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:56:45.559347Z"},"content_sha256":"c5eed672c8a60d20aba13dc14c0c0d585b2a4ce8f5a45fdea0ae2ce4878015ce","schema_version":"1.0","event_id":"sha256:c5eed672c8a60d20aba13dc14c0c0d585b2a4ce8f5a45fdea0ae2ce4878015ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:RL3OY33SEJBYHNKVFRTYI45QOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Riemannian geometry for low-rank matrix completion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.OC"],"primary_cat":"cs.LG","authors_text":"B. Mishra, K. Adithya Apuroop, R. Sepulchre","submitted_at":"2012-11-07T13:49:26Z","abstract_excerpt":"We propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least square cost function. At one level, it illustrates in a novel way how to exploit the versatile framework of optimization on quotient manifold. At another level, our algorithm can be considered as an improved version of LMaFit, the state-of-the-art Gauss-Seidel algorithm. We develop necessary tools needed to perform both first-order and second-orde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1550","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":"2026-05-18T03:41:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FezcVy+moq38Ve/cZ7wfmc4bhRuIUKWiyTjDt20FKbsaBIlt9gdIZgd86aTrY8IHIQdkkJOoXo7zh5YleAjrAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T13:56:45.559705Z"},"content_sha256":"7a4d2b3a098389661b6e99722a2fceb3b69ecc2a9dcc1a211930bd927cac5a41","schema_version":"1.0","event_id":"sha256:7a4d2b3a098389661b6e99722a2fceb3b69ecc2a9dcc1a211930bd927cac5a41"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RL3OY33SEJBYHNKVFRTYI45QOH/bundle.json","state_url":"https://pith.science/pith/RL3OY33SEJBYHNKVFRTYI45QOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RL3OY33SEJBYHNKVFRTYI45QOH/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":"2026-06-02T13:56:45Z","links":{"resolver":"https://pith.science/pith/RL3OY33SEJBYHNKVFRTYI45QOH","bundle":"https://pith.science/pith/RL3OY33SEJBYHNKVFRTYI45QOH/bundle.json","state":"https://pith.science/pith/RL3OY33SEJBYHNKVFRTYI45QOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RL3OY33SEJBYHNKVFRTYI45QOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RL3OY33SEJBYHNKVFRTYI45QOH","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":"b37d4bc35f3ed799bc6d7c1f6ee2afe0cf2337970ec828bb27e5daa8846b495f","cross_cats_sorted":["cs.NA","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-11-07T13:49:26Z","title_canon_sha256":"f98bf077e32ff5907398da06f7e0d4323be1b4ae6d6bc6dfd8ee097186966c29"},"schema_version":"1.0","source":{"id":"1211.1550","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.1550","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"arxiv_version","alias_value":"1211.1550v2","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.1550","created_at":"2026-05-18T03:41:02Z"},{"alias_kind":"pith_short_12","alias_value":"RL3OY33SEJBY","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RL3OY33SEJBYHNKV","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RL3OY33S","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:7a4d2b3a098389661b6e99722a2fceb3b69ecc2a9dcc1a211930bd927cac5a41","target":"graph","created_at":"2026-05-18T03:41:02Z","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 propose a new Riemannian geometry for fixed-rank matrices that is specifically tailored to the low-rank matrix completion problem. Exploiting the degree of freedom of a quotient space, we tune the metric on our search space to the particular least square cost function. At one level, it illustrates in a novel way how to exploit the versatile framework of optimization on quotient manifold. At another level, our algorithm can be considered as an improved version of LMaFit, the state-of-the-art Gauss-Seidel algorithm. We develop necessary tools needed to perform both first-order and second-orde","authors_text":"B. Mishra, K. Adithya Apuroop, R. Sepulchre","cross_cats":["cs.NA","math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-11-07T13:49:26Z","title":"A Riemannian geometry for low-rank matrix completion"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.1550","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:c5eed672c8a60d20aba13dc14c0c0d585b2a4ce8f5a45fdea0ae2ce4878015ce","target":"record","created_at":"2026-05-18T03:41:02Z","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":"b37d4bc35f3ed799bc6d7c1f6ee2afe0cf2337970ec828bb27e5daa8846b495f","cross_cats_sorted":["cs.NA","math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LG","submitted_at":"2012-11-07T13:49:26Z","title_canon_sha256":"f98bf077e32ff5907398da06f7e0d4323be1b4ae6d6bc6dfd8ee097186966c29"},"schema_version":"1.0","source":{"id":"1211.1550","kind":"arxiv","version":2}},"canonical_sha256":"8af6ec6f72224383b5552c678473b071df987937bbe667d9cec0e51a15dee585","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8af6ec6f72224383b5552c678473b071df987937bbe667d9cec0e51a15dee585","first_computed_at":"2026-05-18T03:41:02.277438Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:41:02.277438Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0enCQdV6FMP9BL/NQTP6qLR584BbKhn/FJcywAhDr30bJEl5cAsZ1lMlOJSQKQqvwx3bFZbOkJp7zdv+dwHWDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:41:02.278047Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.1550","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c5eed672c8a60d20aba13dc14c0c0d585b2a4ce8f5a45fdea0ae2ce4878015ce","sha256:7a4d2b3a098389661b6e99722a2fceb3b69ecc2a9dcc1a211930bd927cac5a41"],"state_sha256":"791bcce938ee7cd3c4d91b04dd6b4f0f969d352f62e9434d0dd4885775fbd19b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VOc+x0Ufy5fJTEb81OVl+HLRQB2mTRXzSCgHtNvXR29FoQhVVbrfvRtbg4PNwXz5v0FydTZv7k3rSWMpuRvmCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T13:56:45.561727Z","bundle_sha256":"37c9139923e799252d0cd8ddeaa42cc1fbd9106b807b487cf180edef4d0ff44d"}}