{"paper":{"title":"Estimating eigenvectors and eigenspaces of covariance matrices: Optimal Bounds and Conditions for Consistency","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["cs.NA","math.NA","math.PR","stat.TH"],"primary_cat":"math.ST","authors_text":"Phuc Tran, Van Vu","submitted_at":"2026-07-27T03:25:16Z","abstract_excerpt":"Let $X = [ \\xi_1, \\,\\, \\xi_2,...\\,\\, ,\\xi_d]^\\top$ be a zero-mean random vector of large dimension $d$ ($d \\rightarrow \\infty$) with (hidden) covariance matrix $M = (m_{ij})_{1 \\leq i, j \\leq d},$ where $m_{ij} = m_{ji} = \\textbf{Cov}(\\xi_i, \\xi_j).$ Let $X_1, X_2, \\dots, X_n$ be $n$ iid samples of $X$. Consider the sample covariance matrix $$\\textstyle \\tilde{M} := \\frac{1}{n} \\sum_{i=1}^{n} X_i X_i^\\top.$$ In practice, one frequently uses the eigenvectors and eigenspaces of $\\tilde M$ as estimators for those of $M$. A central task is to provide an error analysis for these estimators.\n  In th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23964","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/2607.23964/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"}