{"paper":{"title":"Number of relevant directions in Principal Component Analysis and Wishart random matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"cond-mat.stat-mech","authors_text":"Pierpaolo Vivo, Satya N. Majumdar","submitted_at":"2011-12-22T17:18:01Z","abstract_excerpt":"We compute analytically, for large $N$, the probability $\\mathcal{P}(N_+,N)$ that a $N\\times N$ Wishart random matrix has $N_+$ eigenvalues exceeding a threshold $N\\zeta$, including its large deviation tails. This probability plays a benchmark role when performing the Principal Component Analysis of a large empirical dataset. We find that $\\mathcal{P}(N_+,N)\\approx\\exp(-\\beta N^2 \\psi_\\zeta(N_+/N))$, where $\\beta$ is the Dyson index of the ensemble and $\\psi_\\zeta(\\kappa)$ is a rate function that we compute explicitly in the full range $0\\leq \\kappa\\leq 1$ and for any $\\zeta$. The rate functio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.5391","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":""},"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"}