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Two estimators of linear functionals of $μ_φ$ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $φ$, first-order error bounds, in discretization step size, on the bias and variance/mean-square error of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $μ_φ$ and a stationary measure of the discretized Markov process. This order is preserved even upon using retractions when exponential maps are unavailable in closed form, thus enhancing p","title":"Sampling and estimation on manifolds using the Langevin diffusion","url":"https://arxiv.org/pdf/2312.14882v3","vendor":"arxiv"},"summary":"Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion with invariant measure $\\text{d}μ_φ\\propto e^{-φ} \\mathrm{dvol}_g $ on a compact Riemannian manifold. Two estimators of linear functionals of $μ_φ$ based on the discretized Markov process are considered: a time-averaging estimator based on a single trajectory and an ensemble-averaging estimator based on multiple independent trajectories. Imposing no restrictions beyond a nominal level of smoothness on $φ$, first-order error bounds, in discretization step size, on the bias and variance/mean-square error of both estimators are derived. The order of error matches the optimal rate in Euclidean and flat spaces, and leads to a first-order bound on distance between the invariant measure $μ_φ$ and a stationary measure of the discretized Markov process. This order is preserved even upon using retractions when exponential maps are unavailable in closed form, thus enhancing p","title":"Sampling and estimation on manifolds using the Langevin diffusion","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/2312.14882v3"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:3aefbe2af6d79e99efd6c6512eb3215262f3569c03569c5b88f9c871cd5cfcf272c1c1aa1035926c8befce74a17f9db6ab45144ca4c604201d33495165bdf207","signer":"crovia.substrate","subject":{"observed_at":"2026-05-03T16:20:12Z","source_collector":"spider:arxiv_retraction","target_id":"https://arxiv.org/pdf/2312.14882v3"},"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":"18197f2e6353ac3e9e6f67c3fbd5900e85b6b86b7d7f7d4823fbb251248739cd","leaf_index":111968,"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":"6053f6f2fd94b0f33f6885d657a11d99b43576db45b3232d9a3a41a9a5ded896","side":"right"},{"sibling":"737e2110d6ea5cce73ba5e633203ce31218bd3f7c31ab34a3c0be1b6cf417bce","side":"right"},{"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_fdef80144a4085d3ab5f9451bf57c3da046214e0963dfea1f19c3f95851fef19"}}