{"paper":{"title":"A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.AI","cs.IT","math.IT","stat.TH"],"primary_cat":"math.ST","authors_text":"Munsik Kim","submitted_at":"2026-06-05T14:43:10Z","abstract_excerpt":"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 tempor"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.07325","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.07325/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}