{"paper":{"title":"Concentration Inequalities and Moment Bounds for Sample Covariance Operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Karim Lounici, Vladimir Koltchinskii","submitted_at":"2014-05-10T20:07:50Z","abstract_excerpt":"Let $X,X_1,\\dots, X_n,\\dots$ be i.i.d. centered Gaussian random variables in a separable Banach space $E$ with covariance operator $\\Sigma:$ $$ \\Sigma:E^{\\ast}\\mapsto E,\\ \\ \\Sigma u = {\\mathbb E}\\langle X,u\\rangle, u\\in E^{\\ast}. $$ The sample covariance operator $\\hat \\Sigma:E^{\\ast}\\mapsto E$ is defined as $$ \\hat \\Sigma u := n^{-1}\\sum_{j=1}^n \\langle X_j,u\\rangle X_j, u\\in E^{\\ast}. $$ The goal of the paper is to obtain concentration inequalities and expectation bounds for the operator norm $\\|\\hat \\Sigma-\\Sigma\\|$ of the deviation of the sample covariance operator from the true covariance"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.2468","kind":"arxiv","version":3},"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"}