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We use Riemannian geometry notions to redefine the optimisation problem of a function on the Euclidean space to a Riemannian manifold with a warped metric, and then find the function's optimum along this manifold. The warped metric chosen for the search domain induces a computational friendly metric-tensor for which optimal search directions associated with geodesic curves on the manifold becomes easier to compute. Performing optimization along geodesics is known to be generally infeasible, yet we show that in this specific manifold we can analytically derive Taylor approximations up to third-order. 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Performing optimization along geodesics is known to be generally infeasible, yet we show that in this specific manifold we can analytically derive Taylor approximations up to third-order. In general these approximations to the geodesic curve will not lie on the manifold, however we construct suitable retraction maps to","title":"Warped geometric information on the optimisation of Euclidean functions","vendor":"arxiv"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-05-03T16:20:12Z","notes":"Spider arxiv_retraction (research) research.arxiv_retraction.v1","object":{"captured_by":"crovia.spider.arxiv_retraction","primary_source_url":"https://arxiv.org/pdf/2308.08305v2"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:e7be0bccc28f20a8ce9b797ad65b1257131e2a2f81232470d0a35f45f42d87f8a158b49a29890c34a258b5a4dd78970645a81be80f8eba889cb3787163efa50d","signer":"crovia.substrate","subject":{"observed_at":"2026-05-03T16:20:12Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2308.08305v2"},"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":"830ba8f5a781cb5fb4884435b51f081477986870b86e496ba71d8648922420cc","leaf_index":111970,"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":"439b20e266f7e20a7a44ed01247661c37769173bb2e967d15e2613912ffa8b5f","side":"right"},{"sibling":"483db0846d3caaa9e0b68d3abcb14880d88611d881b37609c86f6b4ca49c1f12","side":"left"},{"sibling":"aed54e98bfdaedd6cb9f70312f8edbf3ea27b25e7e80f8ebd777b118c655f44e","side":"right"},{"sibling":"9f38bd54d412a7c09de8490ad4d7f5c1f2c2bcfc9c11c180f179c04d4ed0af92","side":"right"},{"sibling":"ad668125d70a43a66c8f0cedc235ad79b00d6fe484a3cb891f2f6d523f9a0fb3","side":"right"},{"sibling":"0507cd3571053c7e209dba71e5e94e1c02f3f4e9f615cc98af787a88c6c4c961","side":"left"},{"sibling":"7cc6ec0331a098790dc04ced8e8abc2b910e0fcce6830c8442fff364ca231498","side":"left"},{"sibling":"0994d51d2178913b0e3ac7346ceed16c5ea436d841a8e7f28a99362fa269bad0","side":"right"},{"sibling":"015d2ce3c58c7a790ad078159267c63cd54f9ae7e20530115a6a423419cefe22","side":"left"},{"sibling":"5bcdb02d4310fc8bfd791789d990714b0a10f44358ffa98c446f1755352c49b8","side":"right"},{"sibling":"cf8897e1feec248f5d05e2cafec4f8c2739e2005227d28b5ccc8cfa11ed66ee4","side":"left"},{"sibling":"a6d6e5bc888edc394757fbba02b675bac6ad0831e36a6f5ebbaa4e5548f25603","side":"right"},{"sibling":"76d7c4385d71ae8104107105be66eddb792e01ed6e493db5a9b4eec5441abb76","side":"left"},{"sibling":"b03f005860bf95148ea89c703a32ca19ab7a2ca71b58bb628ff94a9edb8703b0","side":"left"},{"sibling":"21d1e30b556f553dc939fddc23e6367f0a7755ebe3dd489dc0001cef017beb7f","side":"right"},{"sibling":"f2817ab288b5324fe49770372c7a10f33f7cd11005f8d4c0a730316f5229dc98","side":"left"},{"sibling":"725fac972e772ca0dc598810ea1abc70df472f72d2d6ab8a0baee2b80e5d2f4c","side":"left"},{"sibling":"98fc57dfef8873b512edc8340f7181df57302bb96777625e072235c62d7c5895","side":"right"}]},"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":134292,"merkle_root":"77fc9c28fae777b81da5b495b3115474df6592dfac590333213d3bdf8b94a9b3","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260515T023701Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-05-15T02:37:25Z","sig_algorithm":"ed25519","signature":"68107a834b00b24f5d4501e5ec727445311f132a486567ecc4c72a4e6dff24c8c21f2de3105293353ba5fdbe370d032819af6aa70f694e2e39b6af6737507009","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_22991a97f88f1ec47476322512475575e25f69354acefe23b104df2b4ad9032b"}}