{"paper":{"title":"Sharp Rates of MMD Empirical Estimation with Power Kernels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.OC","math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Francesco Colasanto, Francesco Mattesini, Massimo Fornasier, Matteo Focardi","submitted_at":"2026-05-18T14:49:52Z","abstract_excerpt":"We establish quantitative rates of convergence for the empirical estimation of probability measures by means of the Maximum Mean Discrepancy (MMD) with power kernel $K_q(x,y) = -|x-y|^q$, $q \\in (0,2)$. The resulting discrepancy is the classical energy distance $$\\mathcal E_q^2(\\mu, \\omega) = -\\frac{1}{2}\\iint_{\\mathbb{R}^d \\times \\mathbb{R}^d} |x-y|^q \\, d(\\mu - \\omega)(x)\\, d(\\mu - \\omega)(y),$$ and we ask how fast the best $N$-point empirical approximation $\\inf_{\\mu_N \\in \\mathcal{P}^N}\\mathcal{E}_q(\\mu_N,\\omega)$ decays as $N \\to \\infty$. Given a probability measure $\\omega$ on $\\mathbb{R"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.18497","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/2605.18497/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"}