{"paper":{"title":"Using a Non-Commutative Bernstein Bound to Approximate Some Matrix Algorithms in the Spectral Norm","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Malik Magdon-Ismail","submitted_at":"2011-03-28T19:37:20Z","abstract_excerpt":"We focus on \\emph{row sampling} based approximations for matrix algorithms, in particular matrix multipication, sparse matrix reconstruction, and \\math{\\ell_2} regression. For \\math{\\matA\\in\\R^{m\\times d}} (\\math{m} points in \\math{d\\ll m} dimensions), and appropriate row-sampling probabilities, which typically depend on the norms of the rows of the \\math{m\\times d} left singular matrix of \\math{\\matA} (the \\emph{leverage scores}), we give row-sampling algorithms with linear (up to polylog factors) dependence on the stable rank of \\math{\\matA}. This result is achieved through the application o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.5453","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"}