{"paper":{"title":"Rate of convergence of the smoothed empirical Wasserstein distance","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.IT","math.IT","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Adam Block, Alexander Rakhlin, Yury Polyanskiy, Zeyu Jia","submitted_at":"2022-05-04T15:31:51Z","abstract_excerpt":"Consider an empirical measure $\\mathbb{P}_n$ induced by $n$ iid samples from a $d$-dimensional $K$-subgaussian distribution $\\mathbb{P}$ and let $\\gamma = N(0,\\sigma^2 I_d)$ be the isotropic Gaussian measure. We study the speed of convergence of the smoothed Wasserstein distance $W_2(\\mathbb{P}_n * \\gamma, \\mathbb{P}*\\gamma) = n^{-\\alpha + o(1)}$ with $*$ being the convolution of measures. For $K<\\sigma$ and in any dimension $d\\ge 1$ we show that $\\alpha = {1\\over2}$. For $K>\\sigma$ in dimension $d=1$ we show that the rate is slower and is given by $\\alpha = {(\\sigma^2 + K^2)^2\\over 4 (\\sigma^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2205.02128","kind":"arxiv","version":4},"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/2205.02128/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"}