{"paper":{"title":"First Efficient Convergence for Streaming k-PCA: a Global, Gap-Free, and Near-Optimal Rate","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.LG","math.NA","stat.ML"],"primary_cat":"math.OC","authors_text":"Yuanzhi Li, Zeyuan Allen-Zhu","submitted_at":"2016-07-26T18:46:21Z","abstract_excerpt":"We study streaming principal component analysis (PCA), that is to find, in $O(dk)$ space, the top $k$ eigenvectors of a $d\\times d$ hidden matrix $\\bf \\Sigma$ with online vectors drawn from covariance matrix $\\bf \\Sigma$.\n  We provide $\\textit{global}$ convergence for Oja's algorithm which is popularly used in practice but lacks theoretical understanding for $k>1$. We also provide a modified variant $\\mathsf{Oja}^{++}$ that runs $\\textit{even faster}$ than Oja's. Our results match the information theoretic lower bound in terms of dependency on error, on eigengap, on rank $k$, and on dimension "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07837","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"}