{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:HULU6AXZ6MLKUUP6ORKRUKPZMT","short_pith_number":"pith:HULU6AXZ","canonical_record":{"source":{"id":"1111.5280","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-22T18:41:12Z","cross_cats_sorted":["cs.LG","stat.ML"],"title_canon_sha256":"19888cf26cd1fd92aabdcc8c5b61f68ad210a732983c795f9d6ac76693b30bc9","abstract_canon_sha256":"e89ae52c98675fd0eec665537835386a34a886bd82d033b07d32d4a1e61e3286"},"schema_version":"1.0"},"canonical_sha256":"3d174f02f9f316aa51fe74551a29f964c8ac2ab7bac493b3783f69f9b765dc4c","source":{"kind":"arxiv","id":"1111.5280","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5280","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5280v4","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5280","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"pith_short_12","alias_value":"HULU6AXZ6MLK","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HULU6AXZ6MLKUUP6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HULU6AXZ","created_at":"2026-05-18T12:26:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:HULU6AXZ6MLKUUP6ORKRUKPZMT","target":"record","payload":{"canonical_record":{"source":{"id":"1111.5280","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-22T18:41:12Z","cross_cats_sorted":["cs.LG","stat.ML"],"title_canon_sha256":"19888cf26cd1fd92aabdcc8c5b61f68ad210a732983c795f9d6ac76693b30bc9","abstract_canon_sha256":"e89ae52c98675fd0eec665537835386a34a886bd82d033b07d32d4a1e61e3286"},"schema_version":"1.0"},"canonical_sha256":"3d174f02f9f316aa51fe74551a29f964c8ac2ab7bac493b3783f69f9b765dc4c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:43.275083Z","signature_b64":"cszI6cAWtfiViEFpA8tEq2+IPWcTRDvoOF1zcfw//5jgdU/7N4SLefTmFYgspYl8eGe9/U2rzfGBXQIb49wTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3d174f02f9f316aa51fe74551a29f964c8ac2ab7bac493b3783f69f9b765dc4c","last_reissued_at":"2026-05-18T00:58:43.274436Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:43.274436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.5280","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":"2026-05-18T00:58:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RO+SWraRlmAooNEdGgjOu88+S0tTszfCGYkQcqBJt7jq2LULF7k7+pJndn2/73rncdp7Yxo92g4Ki0DHFfOdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T14:40:40.552785Z"},"content_sha256":"6309474e919e5102dcd1f19e4242c946f075c2cceca53fd45b7fb83ccb093cfc","schema_version":"1.0","event_id":"sha256:6309474e919e5102dcd1f19e4242c946f075c2cceca53fd45b7fb83ccb093cfc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:HULU6AXZ6MLKUUP6ORKRUKPZMT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stochastic gradient descent on Riemannian manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","stat.ML"],"primary_cat":"math.OC","authors_text":"Silvere Bonnabel","submitted_at":"2011-11-22T18:41:12Z","abstract_excerpt":"Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and teste"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5280","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":"2026-05-18T00:58:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7vRT09Y4HgbvXDvJB3xpLe1KpUpVqjw4mi4+CFLYv7NDIfDFd4kEqYwAjjVNYjpGDx3YFpDLg83GfK/eVbRNAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T14:40:40.553124Z"},"content_sha256":"2fac0d39b282993fa7f8db61ea9a834a3253a8b70defb67fbc96dbb198abaa67","schema_version":"1.0","event_id":"sha256:2fac0d39b282993fa7f8db61ea9a834a3253a8b70defb67fbc96dbb198abaa67"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/bundle.json","state_url":"https://pith.science/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/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":"2026-07-04T14:40:40Z","links":{"resolver":"https://pith.science/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT","bundle":"https://pith.science/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/bundle.json","state":"https://pith.science/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HULU6AXZ6MLKUUP6ORKRUKPZMT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:HULU6AXZ6MLKUUP6ORKRUKPZMT","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":"e89ae52c98675fd0eec665537835386a34a886bd82d033b07d32d4a1e61e3286","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-22T18:41:12Z","title_canon_sha256":"19888cf26cd1fd92aabdcc8c5b61f68ad210a732983c795f9d6ac76693b30bc9"},"schema_version":"1.0","source":{"id":"1111.5280","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.5280","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"arxiv_version","alias_value":"1111.5280v4","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.5280","created_at":"2026-05-18T00:58:43Z"},{"alias_kind":"pith_short_12","alias_value":"HULU6AXZ6MLK","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_16","alias_value":"HULU6AXZ6MLKUUP6","created_at":"2026-05-18T12:26:30Z"},{"alias_kind":"pith_short_8","alias_value":"HULU6AXZ","created_at":"2026-05-18T12:26:30Z"}],"graph_snapshots":[{"event_id":"sha256:2fac0d39b282993fa7f8db61ea9a834a3253a8b70defb67fbc96dbb198abaa67","target":"graph","created_at":"2026-05-18T00:58:43Z","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":"Stochastic gradient descent is a simple approach to find the local minima of a cost function whose evaluations are corrupted by noise. In this paper, we develop a procedure extending stochastic gradient descent algorithms to the case where the function is defined on a Riemannian manifold. We prove that, as in the Euclidian case, the gradient descent algorithm converges to a critical point of the cost function. The algorithm has numerous potential applications, and is illustrated here by four examples. In particular a novel gossip algorithm on the set of covariance matrices is derived and teste","authors_text":"Silvere Bonnabel","cross_cats":["cs.LG","stat.ML"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-22T18:41:12Z","title":"Stochastic gradient descent on Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.5280","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":"sha256:6309474e919e5102dcd1f19e4242c946f075c2cceca53fd45b7fb83ccb093cfc","target":"record","created_at":"2026-05-18T00:58:43Z","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":"e89ae52c98675fd0eec665537835386a34a886bd82d033b07d32d4a1e61e3286","cross_cats_sorted":["cs.LG","stat.ML"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-22T18:41:12Z","title_canon_sha256":"19888cf26cd1fd92aabdcc8c5b61f68ad210a732983c795f9d6ac76693b30bc9"},"schema_version":"1.0","source":{"id":"1111.5280","kind":"arxiv","version":4}},"canonical_sha256":"3d174f02f9f316aa51fe74551a29f964c8ac2ab7bac493b3783f69f9b765dc4c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3d174f02f9f316aa51fe74551a29f964c8ac2ab7bac493b3783f69f9b765dc4c","first_computed_at":"2026-05-18T00:58:43.274436Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:43.274436Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cszI6cAWtfiViEFpA8tEq2+IPWcTRDvoOF1zcfw//5jgdU/7N4SLefTmFYgspYl8eGe9/U2rzfGBXQIb49wTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:43.275083Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.5280","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6309474e919e5102dcd1f19e4242c946f075c2cceca53fd45b7fb83ccb093cfc","sha256:2fac0d39b282993fa7f8db61ea9a834a3253a8b70defb67fbc96dbb198abaa67"],"state_sha256":"6b1dc5b03fbcc2ef057efa883d6613249d2f5e834838c0cd3cee729a158a14b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8dzxjmWY5B9GUh+0QxmISNUMVxcmOu7P22I+VyYXi3Ww54k2FX8a7dbVcos2Z0BNUsfCTYqv6ZBVoz2c0bLAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T14:40:40.555052Z","bundle_sha256":"52cbee7e595d18787edebbf29afc557fc61515c31365a203c6732c1a32afe820"}}