{"paper":{"title":"Robust Shift-and-Invert Preconditioning: Faster and More Sample Efficient Algorithms for Eigenvector Computation","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, Praneeth Netrapalli, Sham M. Kakade","submitted_at":"2015-10-29T20:47:27Z","abstract_excerpt":"We provide faster algorithms and improved sample complexities for approximating the top eigenvector of a matrix.\n  Offline Setting: Given an $n \\times d$ matrix $A$, we show how to compute an $\\epsilon$ approximate top eigenvector in time $\\tilde O ( [nnz(A) + \\frac{d \\cdot sr(A)}{gap^2}]\\cdot \\log 1/\\epsilon )$ and $\\tilde O([\\frac{nnz(A)^{3/4} (d \\cdot sr(A))^{1/4}}{\\sqrt{gap}}]\\cdot \\log1/\\epsilon )$. Here $sr(A)$ is the stable rank and $gap$ is the multiplicative eigenvalue gap. By separating the $gap$ dependence from $nnz(A)$ we improve on the classic power and Lanczos methods. We also im"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.08896","kind":"arxiv","version":2},"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"}