{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:GBWFLWFMJQNF7JQGKCRMXZSO27","short_pith_number":"pith:GBWFLWFM","canonical_record":{"source":{"id":"2006.02572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-06-03T23:00:27Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"ba891d201c09c0df1d85623cf0851269f21e6d51d46be420705ca9b6b0bbd894","abstract_canon_sha256":"acb8ee0dc7534d3404fc9300499143eeb38d1f310f4ed0a51845dfd7199f19bd"},"schema_version":"1.0"},"canonical_sha256":"306c55d8ac4c1a5fa60650a2cbe64ed7e1df9496396383a46753ee326487d9d0","source":{"kind":"arxiv","id":"2006.02572","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02572","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02572v2","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02572","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_12","alias_value":"GBWFLWFMJQNF","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_16","alias_value":"GBWFLWFMJQNF7JQG","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_8","alias_value":"GBWFLWFM","created_at":"2026-07-05T01:59:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:GBWFLWFMJQNF7JQGKCRMXZSO27","target":"record","payload":{"canonical_record":{"source":{"id":"2006.02572","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-06-03T23:00:27Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"ba891d201c09c0df1d85623cf0851269f21e6d51d46be420705ca9b6b0bbd894","abstract_canon_sha256":"acb8ee0dc7534d3404fc9300499143eeb38d1f310f4ed0a51845dfd7199f19bd"},"schema_version":"1.0"},"canonical_sha256":"306c55d8ac4c1a5fa60650a2cbe64ed7e1df9496396383a46753ee326487d9d0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:59:12.393150Z","signature_b64":"psl3g7kJC50uC5eWdmkVOXJY4bklRQ0nmOu3gS30lL26CNYnz1TsmTR8znnMo/59gY8NOix8w/RFyCaq8YcfBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"306c55d8ac4c1a5fa60650a2cbe64ed7e1df9496396383a46753ee326487d9d0","last_reissued_at":"2026-07-05T01:59:12.392734Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:59:12.392734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.02572","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":"2026-07-05T01:59:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"owV/JX29azrSKp2mBjhZ6g3lJdRFKpY84j7XTf7z964yOFbXKb6zvUfTFKIpiAfpQ3WPDMR1U6ILQLunHiWMCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T03:37:33.337997Z"},"content_sha256":"03baee7b066ecd904cdb3ee6875bdd9fa6c691a81f97c500a8b2bccbdbf62923","schema_version":"1.0","event_id":"sha256:03baee7b066ecd904cdb3ee6875bdd9fa6c691a81f97c500a8b2bccbdbf62923"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:GBWFLWFMJQNF7JQGKCRMXZSO27","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Boris Muzellec, Gabriel Peyr\\'e, Hicham Janati, Marco Cuturi","submitted_at":"2020-06-03T23:00:27Z","abstract_excerpt":"Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersecti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02572","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/2006.02572/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":"2026-07-05T01:59:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wx3ZV5uF0NrsLta4OX4EcLrzFJUSfl13v2fQUM1PqTf2cZHrXOTSAmjQLsy8qAjmpsSJl33HPBLFRqHFKtSWAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T03:37:33.338582Z"},"content_sha256":"0fc11adbaaa2b627300a1bad352d057ebc1ba2da0e65a8e6bfb4bf0239415fd5","schema_version":"1.0","event_id":"sha256:0fc11adbaaa2b627300a1bad352d057ebc1ba2da0e65a8e6bfb4bf0239415fd5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/bundle.json","state_url":"https://pith.science/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/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":"2026-08-21T03:37:33Z","links":{"resolver":"https://pith.science/pith/GBWFLWFMJQNF7JQGKCRMXZSO27","bundle":"https://pith.science/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/bundle.json","state":"https://pith.science/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GBWFLWFMJQNF7JQGKCRMXZSO27/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:GBWFLWFMJQNF7JQGKCRMXZSO27","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":"acb8ee0dc7534d3404fc9300499143eeb38d1f310f4ed0a51845dfd7199f19bd","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-06-03T23:00:27Z","title_canon_sha256":"ba891d201c09c0df1d85623cf0851269f21e6d51d46be420705ca9b6b0bbd894"},"schema_version":"1.0","source":{"id":"2006.02572","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02572","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02572v2","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02572","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_12","alias_value":"GBWFLWFMJQNF","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_16","alias_value":"GBWFLWFMJQNF7JQG","created_at":"2026-07-05T01:59:12Z"},{"alias_kind":"pith_short_8","alias_value":"GBWFLWFM","created_at":"2026-07-05T01:59:12Z"}],"graph_snapshots":[{"event_id":"sha256:0fc11adbaaa2b627300a1bad352d057ebc1ba2da0e65a8e6bfb4bf0239415fd5","target":"graph","created_at":"2026-07-05T01:59:12Z","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/2006.02572/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Although optimal transport (OT) problems admit closed form solutions in a very few notable cases, e.g. in 1D or between Gaussians, these closed forms have proved extremely fecund for practitioners to define tools inspired from the OT geometry. On the other hand, the numerical resolution of OT problems using entropic regularization has given rise to many applications, but because there are no known closed-form solutions for entropic regularized OT problems, these approaches are mostly algorithmic, not informed by elegant closed forms. In this paper, we propose to fill the void at the intersecti","authors_text":"Boris Muzellec, Gabriel Peyr\\'e, Hicham Janati, Marco Cuturi","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-06-03T23:00:27Z","title":"Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02572","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":"sha256:03baee7b066ecd904cdb3ee6875bdd9fa6c691a81f97c500a8b2bccbdbf62923","target":"record","created_at":"2026-07-05T01:59:12Z","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":"acb8ee0dc7534d3404fc9300499143eeb38d1f310f4ed0a51845dfd7199f19bd","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2020-06-03T23:00:27Z","title_canon_sha256":"ba891d201c09c0df1d85623cf0851269f21e6d51d46be420705ca9b6b0bbd894"},"schema_version":"1.0","source":{"id":"2006.02572","kind":"arxiv","version":2}},"canonical_sha256":"306c55d8ac4c1a5fa60650a2cbe64ed7e1df9496396383a46753ee326487d9d0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"306c55d8ac4c1a5fa60650a2cbe64ed7e1df9496396383a46753ee326487d9d0","first_computed_at":"2026-07-05T01:59:12.392734Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:59:12.392734Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"psl3g7kJC50uC5eWdmkVOXJY4bklRQ0nmOu3gS30lL26CNYnz1TsmTR8znnMo/59gY8NOix8w/RFyCaq8YcfBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:59:12.393150Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.02572","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:03baee7b066ecd904cdb3ee6875bdd9fa6c691a81f97c500a8b2bccbdbf62923","sha256:0fc11adbaaa2b627300a1bad352d057ebc1ba2da0e65a8e6bfb4bf0239415fd5"],"state_sha256":"e72e76d644e7c1e3a12b6772f73341a60a52aed9cc7504ba8b28735f1d6371b6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0rbWvIOMugng2m67zgYFbBHlxwlxsz3rqF+/IwkeVrXGs3b7RJdNkNkyuY4DhNP65M5EGRePP65Uts2JRTaDAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T03:37:33.343263Z","bundle_sha256":"e07af8304fb5d9280942bc3c2e445f8a4fe377bd4289b885b62aac488baa731c"}}