{"paper":{"title":"Exploiting Numerical Sparsity for Efficient Learning : Faster Eigenvector Computation and Regression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG","math.OC"],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Neha Gupta","submitted_at":"2018-11-27T08:22:54Z","abstract_excerpt":"In this paper, we obtain improved running times for regression and top eigenvector computation for numerically sparse matrices. Given a data matrix $A \\in \\mathbb{R}^{n \\times d}$ where every row $a \\in \\mathbb{R}^d$ has $\\|a\\|_2^2 \\leq L$ and numerical sparsity at most $s$, i.e. $\\|a\\|_1^2 / \\|a\\|_2^2 \\leq s$, we provide faster algorithms for these problems in many parameter settings.\n  For top eigenvector computation, we obtain a running time of $\\tilde{O}(nd + r(s + \\sqrt{r s}) / \\mathrm{gap}^2)$ where $\\mathrm{gap} > 0$ is the relative gap between the top two eigenvectors of $A^\\top A$ and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.10866","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"}