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In particular, this yields a linear time algorithm if the gap is at least $1/\\sqrt{n}$ and $k,p,\\tild"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1607.02925","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-07-11T12:47:39Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"a2d70f9f35bb5cf7372adb502d1190ca25b5942148e3f708aa743146dc7d05e6","abstract_canon_sha256":"d0b24a540e47040ddd18c224cfe40016f5b891858ed8df7f480000f8af48af71"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:14.371048Z","signature_b64":"34zHq9X2pBEFwwxDy9yUldSiGW24gzTmtNO8I5LJt1ptuaAg4m/cgdMRMBaQbIPh6vWvMyM5rJyqfWTr5atKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9d1f7795d7de88dad78531e81c698dff28e469ef3e4431248fc66a56a40a180e","last_reissued_at":"2026-05-18T01:11:14.370460Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:14.370460Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $X$. 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