{"as_of":"2026-08-23T17:52:00Z","caps":{"database_statements":6,"inbound":100,"outbound":100},"context_digest":"sha256:507504335907dcccd81dffb98cfcc1c293085b6b4f51051c8eca138c7f905743","coverage":[{"denominator":49,"lane":"reference_resolution","note":"Typed states for the displayed outbound observations.","records_observed":49,"source":"paper_references, paper_reference_links","source_observed_at":"2026-06-26T00:21:04.782902Z","state":"measured"},{"denominator":49,"lane":"standing_notices","note":"One-hop event checks from named stored sources.","records_observed":49,"source":"scholarly_work_events, retraction_status_cache","source_observed_at":"2026-08-23T06:30:58.430688+00:00","state":"measured"},{"denominator":0,"lane":"inbound_itemization","note":"Pith citing papers itemized under the disclosed page cap.","records_observed":0,"source":"paper_references, paper_reference_links","source_observed_at":null,"state":"measured"},{"denominator":1,"lane":"external_citation_measurements","note":"A source-named dated measurement, never combined with another source.","records_observed":0,"source":"cited_works","source_observed_at":null,"state":"measured"}],"external_citation_measurements":[],"inbound":[],"links":{"evidence":"/evidence","html":"/paper/2606.24157/citation-record","integrity":"/paper/2606.24157/integrity","json":"/paper/2606.24157/citation-record.json","paper":"/paper/2606.24157"},"outbound":[{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"It is the equation of motion in Wasserstein space","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":1,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:5095b148975471fad795a46396d234f1d722d29e0e917d9612ceb4e2c23a702f","observation_id":"5bfb8a54-121b-4d7b-bade-72d32df90e69","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":2,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:4f8a7458d34934c31e16b5c3fd7a2a996fd66d336e2a29836e16f9bb836b449d","observation_id":"3fe42c96-65a8-416f-9f43-00c40b660765","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"˙ρ=−gradW F(ρ)","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":3,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:e28b85d0ee710959202ba5c8cfd4f7affa1acbb6897432a2f83db8901007a243","observation_id":"4b45f04f-0a6a-492f-a1f5-5155719ee565","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":4,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:fc790f6519a2bae9c4d71015d0bed8e117b2eaebf64baaf016b7f6e42f62cb8b","observation_id":"cec16f89-3513-45d6-9ac7-13c2105deb00","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(1781).M´ emoire sur la th´ eorie des d´ eblais et des remblais.Histoire de l’Acad´ emie Royale des Sciences de Paris, 666–704","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":5,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:ec7ffea758306dffaaa107ec8722d97d8e7d865595296a62cb4de9584d0e1a59","observation_id":"c719863f-7fc3-4ed9-bc1b-45627befd7ec","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1942},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":6,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:7dc44b1909269db59bf99bed737d431d73ce0a165d276f86f40e41240e20d337","observation_id":"21b7a54a-e917-45bb-a871-1bd47e1a271d","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(1991).Polar factorization and monotone rearrangement of vector-valued functions.Comm","venue":null,"work_id":null,"year":1991},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":7,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:c109d37c7c57fb81486d0bbf3e9a0c95c29470bb3ff7d1bcf2aaa2272da27eda","observation_id":"22f7d7d2-3044-4df2-b3b7-cf1a592f3915","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"Wasserstein distance","venue":null,"work_id":null,"year":1969},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":8,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:48df1a78f13a075b48cd513b908b47934cfd141cf50dced03b61a9e3083429d9","observation_id":"d91cf69c-2eec-4813-bfed-3488ae81d0b2","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1997},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":9,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:654d115c0c92192f6d965f1e47fc6416d2ffd29ca9c64b0277fbc5ca81957967","observation_id":"e66f5472-c4b9-48f7-8bea-858c3554e734","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Brenier, Y","venue":null,"work_id":null,"year":2000},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":10,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:ba1392bf71cfc3f6662c883554907e49915beee541ab4c4bef2dcd5d3b1b7496","observation_id":"3781693a-8fe6-429f-8fb1-157422ea4d08","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(1998).The variational formulation of the Fokker–Planck equation.SIAM J","venue":null,"work_id":null,"year":1998},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":11,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:fdb930eca9c3cb69e6111c7bc80775bad6319beb51e2c412f3f961fa8a9616df","observation_id":"8716893b-bef9-4ddb-9af7-fde58ce6240d","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2001).The geometry of dissipative evolution equations: the porous medium equation.Comm","venue":null,"work_id":null,"year":2001},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":12,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:f6ea4e48d01ac275397d8ce6d078a4c842026673d01e905c27474b493303bf86","observation_id":"e7d28870-efa5-4707-be4a-2fd6f628173a","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2008).Gradient Flows in Metric Spaces and in the Space of Probability Measures.Birkh¨ auser","venue":null,"work_id":null,"year":2008},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":13,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:6e820bc1f193100fbff383b410003f4513d03e99626da3a00cb38ab338ee634c","observation_id":"95458fa3-067c-4c72-b121-2f6b139e92bf","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2003).Topics in Optimal Transportation.AMS","venue":null,"work_id":null,"year":2003},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":14,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:b4c19b85f145a3be1f666e12fa4fee97ea33c588b401c83e73fc551f73fb3bd0","observation_id":"adb4fdce-04c6-452a-b941-093a4f1b01a5","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2015).Optimal Transport for Applied Mathematicians.Birkh¨ auser","venue":null,"work_id":null,"year":2015},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":15,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:d0ba32c978b5b48b525c9ee4af186bff904e60af8fac66793d8d598d2c6b9233","observation_id":"17cfef75-5d73-4839-a84e-a19e52d0e5d9","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Cuturi, M","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":16,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:3b58c1c35acecab07aa3c19a51c7f3c148b567ad2ce866cb981ac73217a61e21","observation_id":"8760b78f-77e0-4a83-bac8-577d8f1f4f8b","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Santambrogio, F","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":17,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:f2c577aa4a93e1dd3b1cadc61750a7fba6e049446ef531b9b721d77e69995005","observation_id":"d4ba500a-a941-4984-8e60-e3fb6935a1d1","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2021).Large- scale Wasserstein gradient flows.NeurIPS","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":18,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:a9fbcab45475da84c9582492e6d791e8b5c6322a8d0283fc527809f8c8fcc3e7","observation_id":"336ed51e-ddbc-41de-acb8-dd59c25349e2","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2023).Normalizing flow neural networks by JKO scheme","venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":19,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:00de355042eb61fc559a11bc0167dbb5e02cd57a656d77a2678b7f370dd60f84","observation_id":"f31a5feb-64db-4553-81ef-effa20c21c80","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2005).Estimation of non-normalized statistical models by score matching","venue":null,"work_id":null,"year":2005},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":20,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:7b2c1d26ba43b36075b7e3954a59c375302f2bdb0c37e62dc90f22a09ed3a57b","observation_id":"4f8ef443-971a-4110-be3d-c0a2da6ddf7f","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2011).A connection between score matching and denoising autoencoders","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":21,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:08f60d8c70e59979423a9b915ae28e612460dd3d9cb97a856bfc9f697110160c","observation_id":"6feb65ff-c71c-4018-adf5-cb374b4a55ca","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Teh, Y","venue":null,"work_id":null,"year":2011},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":22,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:02052173814f9708c85abd0c6f1f3c52918ce1ee12765b71fa6031acb06164ec","observation_id":"5a95df5f-7637-4d4f-b132-a7a3ada9d625","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Ermon, S","venue":null,"work_id":null,"year":2019},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":23,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:fe1f18293b88f630c9dc98fb7c46d4a6a389208244bd17e967ad19ece12cc2a1","observation_id":"d0bbc3a6-4090-42cc-8e95-9f015e2af6ed","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2020).Denoising diffusion probabilistic models.NeurIPS","venue":null,"work_id":null,"year":2020},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":24,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:ca24f2d45dfe8776c605a28f7f398d6ddd8e384c2426963cf89192c4c10dd22b","observation_id":"2f1571b7-7ae3-4f28-bb48-2a1447e96b5b","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2021).Denoising diffusion implicit models.ICLR","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":25,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:64e3255873b751cbaebd8abcaf7bd8075126bc3d5ccbfafb4f46362c6a0a3849","observation_id":"69a24b81-94d0-4951-b803-dfdbb7795db6","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"P., Kumar, A., Ermon, S., & Poole, B","venue":null,"work_id":null,"year":2021},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":26,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:e0ba8b476bc4aad3bb14ed8eaf1512402df42c08e3db71d8cf193902398d2236","observation_id":"eddcdc17-8242-4c3c-9644-85429a508bfc","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(2022).Elucidating the design space of diffusion-based generative models.NeurIPS","venue":null,"work_id":null,"year":2022},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":27,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:a1886ed7ccac6457c3eeaf7e5930b87c55a5687c0c0ed7cdbb3b79dbcc9f0091","observation_id":"f38ff38e-d740-4362-ba23-63559e9ab7f1","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":2023},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":28,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:efb2b52cf728b257c485976b89839290b8f3a6a0eda3d8f2b9e7bf3d81eb9331","observation_id":"a085a474-d254-479e-8d62-aa6bd2bd1f47","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2504.10612","doi":"10.48550/arxiv.2504.10612","metadata_source":"arxiv_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Energy matching: Unifying flow matching and energy-based models for generative modeling","venue":"ArXiv.org","work_id":"d0d065c8-17d7-404d-87f8-af8a3656897d","year":2025},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":29,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:30c7f8f709ccb8cce1db1beb1b2e4e734d6c7004ec00ed148fa821619883f79a","observation_id":"d8394fb2-577f-497c-9682-3f3ca889a93d","resolution":{"observed_at":"2026-07-04T16:39:57.924494Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-23T06:30:58.430688+00:00","source":"crossref"},{"observed_at":"2026-08-23T06:30:53.778098+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":"2510.02300","doi":null,"metadata_source":"arxiv_reference","pith_arxiv_id":null,"snapshot_observed_at":"2026-07-04T16:39:57.925606Z","title":"Equilibrium matching: Generative modeling with implicit energy-based models","venue":null,"work_id":"42702731-dd64-451a-b3af-058139a91036","year":2025},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":30,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:b0895011cc5ee055a1f43112dc82be6a75668932d016045253e79db8fd1ec211","observation_id":"2a5b6730-0fea-406b-b091-ade1c02a4746","resolution":{"observed_at":"2026-07-04T16:39:57.927468Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-23T06:30:58.430688+00:00","source":"crossref"},{"observed_at":"2026-08-23T06:30:53.778098+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"S., Boffi, N","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":31,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:036ac75c907571672ad44b0f3f8c6af8207e270fc0431e55b34aa9c0ab11d1bb","observation_id":"e5b4d206-ff20-4cd1-8abb-24270fb3ae9e","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":{"arxiv_id":"2412.06264","last_updated":"2024-12-09T07:22:38Z","snapshot_observed_at":"2026-08-20T22:22:58.160467Z","submitted_at":"2024-12-09T07:22:38Z","title":"Flow Matching Guide and Code","version":1},"cited_work":{"arxiv_id":"2412.06264","doi":"10.48550/arxiv.2412.06264","metadata_source":"pith","pith_arxiv_id":"2412.06264","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"Flow Matching Guide and Code","venue":"cs.LG","work_id":"2be93143-ab6f-48d6-96d5-3e85d7246f07","year":2024},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":32,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"cited_paper":"/paper/2412.06264","citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:c7adf8f0e92ca3cde161de6f380e228b2d8855266900ab8f563594d310aba17b","observation_id":"d59fe256-3e00-4d87-8473-73bc9673410e","resolution":{"observed_at":"2026-07-04T16:39:57.915199Z","resolver_source":"local_arxiv","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-23T06:30:58.430688+00:00","source":"crossref"},{"observed_at":"2026-07-12T13:49:33.385348+00:00","source":"crossref_status_cache"},{"observed_at":"2026-07-12T13:49:33.385348+00:00","source":"openalex_status_cache"},{"observed_at":"2026-08-23T06:30:53.778098+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":{"arxiv_id":"2509.00336","last_updated":"2025-08-30T03:30:22Z","snapshot_observed_at":"2026-08-17T18:15:38.664885Z","submitted_at":"2025-08-30T03:30:22Z","title":"Are We Really Learning the Score Function? Reinterpreting Diffusion Models Through Wasserstein Gradient Flow Matching","version":1},"cited_work":{"arxiv_id":"2509.00336","doi":"10.48550/arxiv.2509.00336","metadata_source":"arxiv_reference","pith_arxiv_id":"2509.00336","snapshot_observed_at":"2026-08-05T02:28:24.338817Z","title":"URLhttps://arxiv.org/abs/2509.00336","venue":"ArXiv.org","work_id":"6a955648-e1a9-498a-a84f-b854f02fa5f2","year":2025},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":33,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"cited_paper":"/paper/2509.00336","citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:aa8b1aa5224d6d749a52ea9b2b49b54fefc3262dbf604417b1f60a16bebb0dbb","observation_id":"9d5c1e78-779a-438f-8e20-ee874f7c06bf","resolution":{"observed_at":"2026-07-04T16:39:57.925571Z","resolver_source":"arxiv_id","status":"verified_exact"},"standing_notice":{"events":[],"observation":"No event found in the named queried sources as of 2026-08-23T06:30:58.430688+00:00.","reason":null,"source_receipts":[{"observed_at":"2026-08-23T06:30:58.430688+00:00","source":"crossref"},{"observed_at":"2026-08-23T06:30:53.778098+00:00","source":"retraction_watch"}],"state":"measured"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Erives, E","venue":null,"work_id":null,"year":2025},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":34,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:3b7871ee7a205beb68925ef7211f0517f74b18c3088ce6dbf44435e3e3cbd8b4","observation_id":"ad62e94e-0ccb-4b84-82dd-c4368d30ccb3","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1925},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":35,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:ac1613751be873845dc4654eacc914223cf0e242db42696d91658130c8184a96","observation_id":"9f1c3ad4-e07c-43e2-96fb-4537ed5a3e6e","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(1908).Sur la th´ eorie du mouvement brownien.C","venue":null,"work_id":null,"year":1908},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":36,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:70a4e6866f480013b0a7c78f74c306f585612af787b4ed8dc92570ffd0b0d224","observation_id":"9ef570c3-0660-444a-adcf-691d95974f74","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1930},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":37,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:b60196e458ad4368bf8230b77c25aff81a2cbd54a2aea9c419ccd9d7b002d212","observation_id":"a758e43b-7f40-43af-842d-187459056519","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1931},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":38,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:67bc03f71c6042a909bfdcd9c5651e90286614cbe6738aa759a07b05652d0195","observation_id":"02962943-6835-4eff-a764-99685f372087","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"(1951).On stochastic differential equations.Mem","venue":null,"work_id":null,"year":1951},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":39,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:366503f58b726557387f698c02c7daa7899451471c1580e4009b63659a09f727","observation_id":"7f92b414-5f90-4030-a769-7e587a86fcb7","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"& Leibler, R","venue":null,"work_id":null,"year":1951},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":40,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:539479449a5104c9805896ec40b16d72f0ede00daebf148ef92d09592cd14a58","observation_id":"4f18b731-7c8f-4082-956c-d7d2327d97af","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":1966},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":41,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:ca37817aab7d7c03496146a7a0aa11c04b8eec5e6d986fa6eba7bb103f0194ee","observation_id":"3e8ec43d-1a41-40d8-8547-2335beec747e","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"assigning size to sets","venue":null,"work_id":null,"year":1982},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":42,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:9b5233d538dc95e2302b53e49ce2886b405e88e4624cef01f6d0c574ad9424d7","observation_id":"018638cf-1277-4997-9d1f-b83474fa89c0","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"the densityρ(x) being large at some point","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":43,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:2cd70997f6cb767db186cec0eee3ffa32a18012d84edbe5501a8ac29c132e5b5","observation_id":"7a0ecb12-519e-40ed-9ee7-b7f16ead8b06","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"transposing","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":44,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:46f88cf7b7bb2e3b29df4a01611340b2691c42e6910c018a3d3515ea5b19e902","observation_id":"467de97d-0110-4622-a906-4f283f74c08e","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":null,"venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":45,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:29ad66fdaa0397f59c5c72a17197ce4c3b819c216127958aa4620da47feb79dc","observation_id":"04b38457-9bac-4de1-84aa-b8e87d2e5396","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"rolling downhill","venue":null,"work_id":null,"year":1925},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":46,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:67d45b9bcfc03e601adc187db62b557de76b4fa201164e4ac781e1d02adba1e6","observation_id":"79c7da21-d531-4804-99c6-a247821eb9d2","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"gradient direction","venue":null,"work_id":null,"year":1982},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":47,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:fb320155357f8a3bdc2cdfe23a9a5495c4e693d75b0ea19b470d37ec55ea1970","observation_id":"68232b33-b1ee-4429-8d8e-ef46359ff57d","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"It is the standard tool for proving stability","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":48,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:680c02203cf12a406f8d94c202762ca91decff8ea599a0717e0e90c90c695b8e","observation_id":"2e2e6594-a455-4301-aac3-14acdea48e37","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}},{"citation":{"cited_paper":null,"cited_work":{"arxiv_id":null,"doi":null,"metadata_source":null,"pith_arxiv_id":null,"snapshot_observed_at":"2026-06-26T00:21:04.782902Z","title":"gradient equals zero","venue":null,"work_id":null,"year":null},"citing_paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space","version":1},"reference_index":49,"source":"pdf_text","source_observed_at":"2026-06-26T00:21:04.782902Z"},"links":{"citing_paper":"/paper/2606.24157"},"observation_digest":"sha256:1a1461826cb900f6a7e81fd99a1ee78f485c309439069efa74be86409f47089b","observation_id":"fd8c9f89-cb4c-46e0-8b6b-446d7488a05a","resolution":{"observed_at":"2026-06-26T00:21:04.782902Z","resolver_source":null,"status":"unresolved"},"standing_notice":{"events":[],"reason":"canonical_work_link_unavailable","source_receipts":[],"state":"unavailable"}}],"paper":{"arxiv_id":"2606.24157","last_updated":"2026-06-23T05:25:24Z","latest_version":1,"primary_category":"cs.AI","snapshot_observed_at":"2026-08-14T10:10:53.191092Z","submitted_at":"2026-06-23T05:25:24Z","title":"The Geometry Behind Diffusion and Flow Matching: Gradient Flows and Geodesics in Wasserstein Space"},"reference_resolution":{"displayed":49,"state_counts":{"malformed_identifier":0,"metadata_mismatch":0,"parse_uncertain":0,"unresolved":45,"verified_exact":4,"verified_fuzzy":0},"total_outbound_references":49},"refusal":"A citation records a reference. It does not transfer a finding from one paper to another.","schema":"pith.paper-citation-record.v1","standing_sources":[{"observed_at":"2026-08-23T06:30:58.430688+00:00","source":"crossref"},{"observed_at":"2026-08-23T06:30:53.778098+00:00","source":"retraction_watch"}],"thesis":"As of 23 August 2026, this Paper Citation Record lists 49 of 49 outbound references and 0 inbound Pith citation observations for arXiv:2606.24157."}