{"paper":{"title":"Faster Eigenvector Computation via Shift-and-Invert Preconditioning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.NA","math.OC"],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Cameron Musco, Chi Jin, Dan Garber, Elad Hazan, Praneeth Netrapalli, Sham M. Kakade","submitted_at":"2016-05-26T03:53:00Z","abstract_excerpt":"We give faster algorithms and improved sample complexities for estimating the top eigenvector of a matrix $\\Sigma$ -- i.e. computing a unit vector $x$ such that $x^T \\Sigma x \\ge (1-\\epsilon)\\lambda_1(\\Sigma)$:\n  Offline Eigenvector Estimation: Given an explicit $A \\in \\mathbb{R}^{n \\times d}$ with $\\Sigma = A^TA$, we show how to compute an $\\epsilon$ approximate top eigenvector in time $\\tilde O([nnz(A) + \\frac{d*sr(A)}{gap^2} ]* \\log 1/\\epsilon )$ and $\\tilde O([\\frac{nnz(A)^{3/4} (d*sr(A))^{1/4}}{\\sqrt{gap}} ] * \\log 1/\\epsilon )$. Here $nnz(A)$ is the number of nonzeros in $A$, $sr(A)$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.08754","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"}