{"_canonicalization":{"envelope_id":"axm_ + sha256(envelope minus {signature, axiom_id, anchors})","envelope_signature":"ed25519(envelope minus {signature, axiom_id})","json":"sort_keys=True, separators=(',',':'), ensure_ascii=False, allow_nan=False, utf-8","leaf_hash":"sha256(0x00 || canonical_json(envelope_full))","seal_signature":"ed25519(seal minus {signature, sig_algorithm})"},"axiom_id":"axm_377f57dce14f4bacc417b265db6146404857220c7d9f706202b215eaf71e3b0b","bitcoin_anchor":{"bitcoin_attestations":[],"calendar_attestations":[],"ots_url":"","stamped_at":"","status":"pending_next_stamp"},"envelope":{"anchors":[{"chain":"crovia.axiom_graph","height":0,"merkle_proof":"spider_arxiv_retraction_v1","root_at_anchor":"spider_arxiv_retraction_v1"}],"axiom_id":"axm_377f57dce14f4bacc417b265db6146404857220c7d9f706202b215eaf71e3b0b","axiom_type":"AX.OBS","body":{"axiom_subtype":"research.arxiv_retraction.v1","category":"research","fingerprint":"f1752bf18bc405e8d214efb56d42500e31f29ef849818905810662a08291c3df","published":"2026-06-10T14:17:08Z","receipt_hash":"f1752bf18bc405e8d214efb56d42500e31f29ef849818905810662a08291c3df","schema":"spider.research.arxiv_retraction.v1","spider":"arxiv_retraction","spider_record":{"axiom_subtype":"research.arxiv_retraction.v1","category":"research","decision_hint":"POSITIVE","envelope_target":"AX.OBS","fingerprint":"f1752bf18bc405e8d214efb56d42500e31f29ef849818905810662a08291c3df","observed_at":"2026-06-11T05:51:56.385100Z","parent_run_hash":"ec8ea61f6eac986358c5e74a80da5b6df751080c4a426c51856e98c2e237168a","published":"2026-06-10T14:17:08Z","runtime_version":"0.1.0","schema":"spider.research.arxiv_retraction.v1","source_status":200,"source_url":"http://export.arxiv.org/api/query?search_query=cat:cs.LG+AND+%28abs:withdrawn+OR+abs:retracted%29&max_results=20&sortBy=submittedDate&sortOrder=descending","spider":"arxiv_retraction","summary_excerpt":"Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely elim","title":"A Riemannian Approach to Low-Rank Optimal Transport","url":"https://arxiv.org/pdf/2606.12120v1","vendor":"arxiv"},"summary":"Low-rank optimal transport (OT) mitigates the quadratic scaling of classical solvers, yet existing approaches rely heavily on first-order mirror-descent updates that require careful hyperparameter tuning and ignore the optimization landscape's curvature. To address these limitations, we propose a unified Riemannian geometric framework for low-rank OT, modeling balanced and unbalanced rank-$r$ positive factored couplings as novel smooth embedded submanifolds of the positive orthant. By equipping these manifolds with the Fisher-Rao product metric, we derive tractable formulations for Riemannian projectors, retractions, and Hessian-vector products. Our cost-agnostic framework seamlessly extends to linear OT, Gromov-Wasserstein (GW), fused GW, and their unbalanced counterparts. For balanced OT, our geometric ingredients are computed via efficient conjugate-gradient and iterative Bregman updates. For the unbalanced OT, our operations elegantly reduce to closed-form scalings, completely elim","title":"A Riemannian Approach to Low-Rank Optimal Transport","vendor":"arxiv"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-11T05:51:56Z","notes":"Spider arxiv_retraction (research) research.arxiv_retraction.v1","object":{"captured_by":"crovia.spider.arxiv_retraction","primary_source_url":"https://arxiv.org/pdf/2606.12120v1"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:5bb4d2bf5cae75848a23553582b592aa11c510a144396bde968a0f2752f4a9c55429f9ba0870121228adb3efe2c546a27aa265ee8ab7d8700a70f64b3a2cbd00","signer":"crovia.substrate","subject":{"observed_at":"2026-06-11T05:51:56Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2606.12120v1"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/research/arxiv_retraction_v1.jsonl\",\"source_seal_merkle_root\":\"spider_arxiv_retraction_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"08bf7a4ca5afe4de0e366186e53eeb443b66071b2fad2043742a4b3e82b9f471","leaf_index":227658,"ledger_path":"/opt/crovia/substrate/axiom_ledger.jsonl"},"merkle_proof":{"hash_alg":"sha256","leaf_prefix":"0x00","node_prefix":"0x01","odd_leaf_rule":"duplicate_last","path":[{"sibling":"ab1b806b1d894d9b9dac58b64b3f60d8a2481ce1cbfb399797c92b3279191d8c","side":"right"},{"sibling":"2d2c8627b3dc0d3c1aa5ccf60788e5b4a9b5e91275fb74c69ff911e17921d779","side":"left"},{"sibling":"b71d4c340a8e8819abfddfb35027cfaa7968f38447a8d93a13663e90c3b1559a","side":"right"},{"sibling":"3b0de66eb5c61f42ee10b9a718cbf86bbc501c01bc191a1a3b649d77df1f9809","side":"left"},{"sibling":"1d9f7cdfaf3fe2cec376523c3652e3f6574ee519bd77580fdf3463d746e6e476","side":"right"},{"sibling":"15c3a07265b83be6930dd7dde6068f607c9dbd2028fea30cabeef6f365717e1d","side":"right"},{"sibling":"fc3fb9294341b82324d688203c5c7dbbef6d9c86c6fccbb04493929ba198c2c1","side":"left"},{"sibling":"8a3e5c7ea49dd66cd71ed0760cc762c1e6d233215488abe34cb40cd8d01bc326","side":"right"},{"sibling":"f76de2250b6d1132375cb2a9c157faf5c22b74a7ca5c8df342d778d0e08e1216","side":"left"},{"sibling":"04e399458c5b36988cae0bf1c6dbe1b01349003b15cb5aa43f95c55acffe4ec3","side":"right"},{"sibling":"1383228337d54218bd8e5563aebb0b0dfe15c5269e3d5138e64c261d6130a88b","side":"right"},{"sibling":"57cb49c192550231071a0bf53a0821da2f79c585ec6c8d0fc76cebd62ccd78b2","side":"left"},{"sibling":"cdb58f86163046d3b15f857b03372ec75e1ad9ea4548e086793d528b9eed364d","side":"left"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"6cea4964f32722eb370847c2f7c9d6a9f0622c239538b07e6815a59d6fd8d49c","side":"right"},{"sibling":"c300cf0154c136afc09b1702a0be98f4ba5b6dc5cf57e8cc714ec1eaf4196eff","side":"left"},{"sibling":"d841ad93efda0869e5eb97678f348f03f5caab4353e05ff4bf18f47fb945b822","side":"left"}]},"schema":"crovia.axiom_proof.v1","seal":{"first_collector_run_id":"","first_receipt_hash":"","jsonl_path":"/opt/crovia/substrate/axiom_ledger.jsonl","key_id":"430895f101d38164","last_collector_run_id":"","last_receipt_hash":"","leaf_count":228173,"merkle_root":"7e416202c0bfd759bd2eea4236713b403993d99793fe8badb5065040080bece3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260611T143708Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-11T21:59:35Z","sig_algorithm":"ed25519","signature":"231c80024bc3982dd493c45b31af95097e97aabc6d712a4e5bad7d0cbdd3c08e01ff395b0f8e72754bac97016e0cd0eed88b8a13cb71edbbcb9b6d72c10a7b03","signer_version":"1.1.0"},"trust_root":{"key_id":"430895f101d38164","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","signature_algorithm":"ed25519","url":"/registry/canon/TRUST_ROOT.md"},"verifier":{"spec":"/registry/canon/AXIOM_RECEIPT_v1.md","url":"/v/axm_377f57dce14f4bacc417b265db6146404857220c7d9f706202b215eaf71e3b0b"}}