{"paper":{"title":"Efficient Estimation of Linear Functionals of Principal Components","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Matthias L\\\"offler, Richard Nickl, Vladimir Koltchinskii","submitted_at":"2017-08-25T08:08:39Z","abstract_excerpt":"We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_1,\\dots, X_n$ in a separable Hilbert space $\\mathbb{H}$ with unknown covariance operator $\\Sigma.$ The complexity of the problem is characterized by its effective rank ${\\bf r}(\\Sigma):= \\frac{{\\rm tr}(\\Sigma)}{\\|\\Sigma\\|},$ where ${\\rm tr}(\\Sigma)$ denotes the trace of $\\Sigma$ and $\\|\\Sigma\\|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $\\Sigma.$ Under the assumption that ${\\bf r}(\\Sigma)=o(n),$ we establish the a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.07642","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":""},"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"}