{"paper":{"title":"Frequent Directions : Simple and Deterministic Matrix Sketching","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"David P. Woodruff, Edo Liberty, Jeff M. Phillips, Mina Ghashami","submitted_at":"2015-01-08T02:32:32Z","abstract_excerpt":"We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix $A \\in R^{n \\times d}$ one row at a time. It performed $O(d \\times \\ell)$ operations per row and maintains a sketch matrix $B \\in R^{\\ell \\times d}$ such that for any $k < \\ell$\n  $\\|A^TA - B^TB \\|_2 \\leq \\|A - A_k\\|_F^2 / (\\ell-k)$ and $\\|A - \\pi_{B_k}(A)\\|_F^2 \\leq \\big(1 + \\frac{k}{\\ell-k}\\big) \\|A-A_k\\|_F^2 $ .\n  Here, $A_k$ stands for the minimizer of $\\|A - A_k\\|_F$ over all rank $k$ matrices (similarly $B_k$) and $\\pi_{B"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01711","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"}