{"paper":{"title":"Minimax Estimation of Kernel Mean Embeddings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Bharath Sriperumbudur, Ilya Tolstikhin, Krikamol Muandet","submitted_at":"2016-02-13T17:53:48Z","abstract_excerpt":"In this paper, we study the minimax estimation of the Bochner integral $$\\mu_k(P):=\\int_{\\mathcal{X}} k(\\cdot,x)\\,dP(x),$$ also called as the kernel mean embedding, based on random samples drawn i.i.d.~from $P$, where $k:\\mathcal{X}\\times\\mathcal{X}\\rightarrow\\mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\\hat{\\theta}_n$ of $\\mu_k(P)$ are studied in the literature wherein all of them satisfy $\\bigl\\| \\hat{\\theta}_n-\\mu_k(P)\\bigr\\|_{\\mathcal{H}_k}=O_P(n^{-1/2})$ with $\\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. T"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04361","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}