{"paper":{"title":"Faster Low-rank Approximation using Adaptive Gap-based Preconditioning","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alon Gonen, Shai Shalev-Shwartz","submitted_at":"2016-07-11T12:47:39Z","abstract_excerpt":"We propose a method for rank $k$ approximation to a given input matrix $X \\in \\mathbb{R}^{d \\times n}$ which runs in time \\[ \\tilde{O} \\left(d ~\\cdot~ \\min\\left\\{n + \\tilde{sr}(X) \\,G^{-2}_{k,p+1}\\ ,\\ n^{3/4}\\, \\tilde{sr}(X)^{1/4} \\,G^{-1/2}_{k,p+1} \\right\\} ~\\cdot~ \\text{poly}(p)\\right) ~, \\] where $p>k$, $\\tilde{sr}(X)$ is related to stable rank of $X$, and $G_{k,p+1} = \\frac{\\sigma_k-\\sigma_p}{\\sigma_k}$ is the multiplicative gap between the $k$-th and the $(p+1)$-th singular values of $X$. In particular, this yields a linear time algorithm if the gap is at least $1/\\sqrt{n}$ and $k,p,\\tild"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.02925","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"}