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Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $\\gamma_d(k+1)/(k+1+\\gamma_d)$, $\\gamma_d=\\min(1/d,1/2)$ ($M$ the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable $W_2$-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are n","title":"A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space","url":"https://arxiv.org/abs/2606.07325","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.07325v1 Announce Type: cross \nAbstract: We study the minimax rate of estimating a future value $\\mu_{t_n+h}$ of a curve $t\\mapsto\\mu_t$ in the $2$-Wasserstein space $\\mathcal{P}_2(\\mathbb{R}^d)$ from finitely many noisy snapshots of its past, under an adiabatic bound $\\|\\nabla_t^k v\\|\\le\\varepsilon$ on the $k$-th covariant derivative of the velocity field. Our central result is a unified temporal-spatial minimax lower bound: over regular, locally transport-rich subclasses, every estimator incurs $W_2$-risk with $M$-exponent $\\gamma_d(k+1)/(k+1+\\gamma_d)$, $\\gamma_d=\\min(1/d,1/2)$ ($M$ the total sample size). It follows from a temporal-to-spatial reduction: the smoothness budget defines a reachable $W_2$-ball into which a transport packing is embedded along the time axis, and the information of the entire snapshot experiment is controlled by a Fano argument -- the spatial packing is classical, but its smoothness-admissible temporal embedding and the full-window analysis are n","title":"A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space","vendor":"arxiv_cs_ai"},"confidence":{"method":"deterministic"},"decision":"POSITIVE","issued_at":"2026-06-08T04:44:02Z","notes":"Spider vendor_press (news) news.vendor_press.v1","object":{"captured_by":"crovia.spider.vendor_press","primary_source_url":"https://arxiv.org/abs/2606.07325"},"predecessors":[],"schema":"crovia.axiom.v1","signature":"ed25519:981e2e19f5e1d4485ead292b48c64b424356a8fabed1e641739ea3c0a60bea5909d7e2bdbe892a25fa70a1762176d05e22851d24934cd3d091c1a25cf07a000a","signer":"crovia.substrate","subject":{"observed_at":"2026-06-08T04:44:02Z","source_collector":"spider:vendor_press","target_id":"https://arxiv.org/abs/2606.07325"},"tsa":{"authority":"crovia.substrate.bootstrap","rfc3161_token":"{\"kind\":\"crovia.bootstrap.tsa\",\"source_jsonl\":\"/opt/crovia/spider/data/news/vendor_press_v1.jsonl\",\"source_seal_merkle_root\":\"spider_vendor_press_v1\",\"upgrade_path\":\"Sessione H \\u2014 OpenTimestamps weekly anchor\"}"},"zk_mode":"clear","zk_proof":null},"ledger":{"leaf_hash":"76f94282e990736a32932ee194595ad24c446d4c12c20e6a1838ecfb6024407b","leaf_index":223780,"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":"bff28f8e8e1126517ce788fe8f029d96d40c717ea1122dbdc816bcdf88719710","side":"right"},{"sibling":"43712e7bbc55b35155ed5e5ae8b12ee63ae5b3ca42ade2466b4aa01e617ae869","side":"right"},{"sibling":"0592e26c0b9a9ab2df83670c0649d90fbe976f94d5ce8d38f02d9d22d476f802","side":"left"},{"sibling":"ea38b0a129872a7e71128d4ede7c7867b1cee5f3b9bed51b9cdda3bf51ee6a53","side":"right"},{"sibling":"06aad92151a4f8b0958774fda4cf8910cda0f88a34bc553508b860d7fd47e885","side":"right"},{"sibling":"77f82ea2f7dc70f1c08e2ec881f2aae96b37e58b007e5808f4ac080d488aabe8","side":"left"},{"sibling":"4e304d5d7910cf045b4d600075b2e1b6a8e413ccb358cfc3bad788ceebcc2de9","side":"right"},{"sibling":"4fcf914698d33c1c26df7cb3719b9425a51cc354d8550fca9c3f04fdd2314f2a","side":"right"},{"sibling":"fad4d9627e4b025e7840b4e896082fb8291cbf1a4c05653b3a789c8a4205d4b5","side":"right"},{"sibling":"e8dcea313a54920d83e4f72d5a4223f986c719f264241d171c4712efaaf1fc54","side":"left"},{"sibling":"5480e1ea31f4744f9bd7c4261771fe51f2cdb01e705cc17320bfc202d935ca12","side":"right"},{"sibling":"24fdc29d461691aedb6fa920758206b5bb43851f477ef7a04c34aaed84b8971b","side":"left"},{"sibling":"036922da4e1e2c46d948f070454bfad299b7406fb00735ea9d8bd1e687f5f445","side":"right"},{"sibling":"533d82482604463aa4a281b9d8b85917b383b7c5f924b2b494039524c55e8797","side":"left"},{"sibling":"b2590791b920ca2a4ed39de126d2c0b1a10d9e7e62f572f12425f214e767b6e1","side":"left"},{"sibling":"87c6b850dfec08ac35a693d9db3a3315250a68adb1cfab9b1015f212b63b15bd","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":224761,"merkle_root":"e9f7b49b652e869ab97ffba9c5a31356b2d0e3dc5d00bb28944adf737c46b1e7","public_key_hex":"cf742e26f75669dc673cb5c0786a1ae23ae8ca19c347317192ce40c28a7ff25c","run_id":"hourly_json_retrofit_20260609T103805Z","schema":"crovia.seal.v1","seal_family_version":"crovia-seal-family/1","seal_kind":"substrate_batch","sealed_at":"2026-06-09T14:15:34Z","sig_algorithm":"ed25519","signature":"8ad8076fb12c8e486ae1d1559a9a7ba8e2ee996a9ad3d8ba7bcdbdbd88ab3a15bcb429707aca6d3e9d8b97e2ba755b3dcc77b1abb6601ccb829842719a6fb30d","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_027febe5e82a39258fe4d063f59640c83d23dc4d4b42208c6ff547c8da532852"}}