{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:ISFANBSQCLGN63NHHKXKR33NHW","short_pith_number":"pith:ISFANBSQ","canonical_record":{"source":{"id":"2110.10464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-10-20T10:03:06Z","cross_cats_sorted":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"title_canon_sha256":"c77f572c55aa3c13a598dd0863dac8e71d7f498211f14773523732901aa1c991","abstract_canon_sha256":"cf0de21610f70d0dd46940660ad56c2e8943da72f65c31bff589e35bc6acf531"},"schema_version":"1.0"},"canonical_sha256":"448a06865012ccdf6da73aaea8ef6d3dacb7f4ac72233b2a3344cd3d273dadb0","source":{"kind":"arxiv","id":"2110.10464","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.10464","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"arxiv_version","alias_value":"2110.10464v2","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.10464","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_12","alias_value":"ISFANBSQCLGN","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_16","alias_value":"ISFANBSQCLGN63NH","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_8","alias_value":"ISFANBSQ","created_at":"2026-07-05T06:18:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:ISFANBSQCLGN63NHHKXKR33NHW","target":"record","payload":{"canonical_record":{"source":{"id":"2110.10464","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-10-20T10:03:06Z","cross_cats_sorted":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"title_canon_sha256":"c77f572c55aa3c13a598dd0863dac8e71d7f498211f14773523732901aa1c991","abstract_canon_sha256":"cf0de21610f70d0dd46940660ad56c2e8943da72f65c31bff589e35bc6acf531"},"schema_version":"1.0"},"canonical_sha256":"448a06865012ccdf6da73aaea8ef6d3dacb7f4ac72233b2a3344cd3d273dadb0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:18:58.250284Z","signature_b64":"RSb7sxHqLETUpJ26NqvkXR/Qw6Fy9Qx/0qllIsdade5wIsdQc0X6Qtufzev0c/lK4jm1eg0OfPIbBJFMONb1AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"448a06865012ccdf6da73aaea8ef6d3dacb7f4ac72233b2a3344cd3d273dadb0","last_reissued_at":"2026-07-05T06:18:58.249849Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:18:58.249849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2110.10464","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":"2026-07-05T06:18:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mpoGQSYeef8pgp8H4w2Tf0xJH2sGIvg3aDswyI1GZPbXI39J6OniYaHArgY4RMhMQJ7e5qNTpgLqLuwNZd93Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T03:55:42.589566Z"},"content_sha256":"3b0bea5ce69bdfed128c17d2a2127b4b3320d8f7243e2e7447ddf6bf3b6dc2aa","schema_version":"1.0","event_id":"sha256:3b0bea5ce69bdfed128c17d2a2127b4b3320d8f7243e2e7447ddf6bf3b6dc2aa"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:ISFANBSQCLGN63NHHKXKR33NHW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"primary_cat":"math.FA","authors_text":"Andi Han, Bamdev Mishra, Junbin Gao, Pratik Jawanpuria","submitted_at":"2021-10-20T10:03:06Z","abstract_excerpt":"Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\\mathbf{M}$ such that when $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.10464","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2110.10464/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T06:18:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UpjuuOB4RJLN1muKooLRZt5DH6cgxJIMusci+fEb/D79vuuenXW+sjYHpjU9nSf7F9yP7e1gZ86h7kxwz1GFAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T03:55:42.590097Z"},"content_sha256":"d35b4558cca676c4da5a383bb77165aaa151bf0950c57709e244d2846027efc8","schema_version":"1.0","event_id":"sha256:d35b4558cca676c4da5a383bb77165aaa151bf0950c57709e244d2846027efc8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ISFANBSQCLGN63NHHKXKR33NHW/bundle.json","state_url":"https://pith.science/pith/ISFANBSQCLGN63NHHKXKR33NHW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ISFANBSQCLGN63NHHKXKR33NHW/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":"2026-08-16T03:55:42Z","links":{"resolver":"https://pith.science/pith/ISFANBSQCLGN63NHHKXKR33NHW","bundle":"https://pith.science/pith/ISFANBSQCLGN63NHHKXKR33NHW/bundle.json","state":"https://pith.science/pith/ISFANBSQCLGN63NHHKXKR33NHW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ISFANBSQCLGN63NHHKXKR33NHW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:ISFANBSQCLGN63NHHKXKR33NHW","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":"cf0de21610f70d0dd46940660ad56c2e8943da72f65c31bff589e35bc6acf531","cross_cats_sorted":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-10-20T10:03:06Z","title_canon_sha256":"c77f572c55aa3c13a598dd0863dac8e71d7f498211f14773523732901aa1c991"},"schema_version":"1.0","source":{"id":"2110.10464","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.10464","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"arxiv_version","alias_value":"2110.10464v2","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.10464","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_12","alias_value":"ISFANBSQCLGN","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_16","alias_value":"ISFANBSQCLGN63NH","created_at":"2026-07-05T06:18:58Z"},{"alias_kind":"pith_short_8","alias_value":"ISFANBSQ","created_at":"2026-07-05T06:18:58Z"}],"graph_snapshots":[{"event_id":"sha256:d35b4558cca676c4da5a383bb77165aaa151bf0950c57709e244d2846027efc8","target":"graph","created_at":"2026-07-05T06:18:58Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2110.10464/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Learning with symmetric positive definite (SPD) matrices has many applications in machine learning. Consequently, understanding the Riemannian geometry of SPD matrices has attracted much attention lately. A particular Riemannian geometry of interest is the recently proposed Bures-Wasserstein (BW) geometry which builds on the Wasserstein distance between the Gaussian densities. In this paper, we propose a novel generalization of the BW geometry, which we call the GBW geometry. The proposed generalization is parameterized by a symmetric positive definite matrix $\\mathbf{M}$ such that when $\\math","authors_text":"Andi Han, Bamdev Mishra, Junbin Gao, Pratik Jawanpuria","cross_cats":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-10-20T10:03:06Z","title":"Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.10464","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":"sha256:3b0bea5ce69bdfed128c17d2a2127b4b3320d8f7243e2e7447ddf6bf3b6dc2aa","target":"record","created_at":"2026-07-05T06:18:58Z","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":"cf0de21610f70d0dd46940660ad56c2e8943da72f65c31bff589e35bc6acf531","cross_cats_sorted":["math.DG","math.OC","math.ST","stat.ML","stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-10-20T10:03:06Z","title_canon_sha256":"c77f572c55aa3c13a598dd0863dac8e71d7f498211f14773523732901aa1c991"},"schema_version":"1.0","source":{"id":"2110.10464","kind":"arxiv","version":2}},"canonical_sha256":"448a06865012ccdf6da73aaea8ef6d3dacb7f4ac72233b2a3344cd3d273dadb0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"448a06865012ccdf6da73aaea8ef6d3dacb7f4ac72233b2a3344cd3d273dadb0","first_computed_at":"2026-07-05T06:18:58.249849Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:18:58.249849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RSb7sxHqLETUpJ26NqvkXR/Qw6Fy9Qx/0qllIsdade5wIsdQc0X6Qtufzev0c/lK4jm1eg0OfPIbBJFMONb1AA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:18:58.250284Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.10464","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b0bea5ce69bdfed128c17d2a2127b4b3320d8f7243e2e7447ddf6bf3b6dc2aa","sha256:d35b4558cca676c4da5a383bb77165aaa151bf0950c57709e244d2846027efc8"],"state_sha256":"401325771c4a09700b61f0924616b7ce1b8de938eeb005a97a3b8bf872776b6d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FSYcd+WwPEqQSnFkhw5jMNbzsE60dSiUWnnMz0+pU9IrGZyLAjpzwy2W6O358fPYvLXiIdgxz1ZILCC33W0TCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T03:55:42.593926Z","bundle_sha256":"144b273e340dca71b7b5f17f4e66a9d01d15e2bc85a5e8639131e06a56323094"}}