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Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of $\\R^{D}$. At the bare minimum, our method requires only a noiseless sample set of the cost function $(\\x_{i}, y_{i})\\in {\\mathcal{M}} \\times \\mathbb{R}$ and the intrinsic dimension of the manifold $\\mathcal{M}$. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components e","title":"Manifold Free Riemannian Optimization","url":"https://arxiv.org/pdf/2209.03269v1","vendor":"arxiv"},"summary":"Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\\mathcal{M}$. Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of $\\R^{D}$. At the bare minimum, our method requires only a noiseless sample set of the cost function $(\\x_{i}, y_{i})\\in {\\mathcal{M}} \\times \\mathbb{R}$ and the intrinsic dimension of the manifold $\\mathcal{M}$. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components e","title":"Manifold Free Riemannian Optimization","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/2209.03269v1"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:a385c8c01dfa4d03556816d51cc8e457b240ac96cb76621c5dd85478a7cf811bf0a03995e6dbc63bd5c6e105745bc60037545cb2140fd0483caac69209e8650b","signer":"crovia.substrate","subject":{"observed_at":"2026-05-03T16:20:12Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2209.03269v1"},"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":"0db1dc0e21f403f78a35b9106a0dbeb9122a417fb4b539ceaab91b7b24b8ee16","leaf_index":111972,"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":"8fcf71cd5612aace134920326e6c297c8b9324e78b2a65b971be81a2f6fa7bd3","side":"right"},{"sibling":"32b76ac9ccdd507ddade90599b175f92890981fe062e330b13944f01b415c7f4","side":"right"},{"sibling":"73eae0107b142a12c04663dff12b884372bb82352fcccde16c0ae7c02e974a63","side":"left"},{"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_f77ca7e7dcc65e01ac988bb347724b80825e8c87a8ab463ea66f8a01d6c23f78"}}