{"paper":{"title":"Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA","math.OC","stat.ML"],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Micha{\\l} Derezi\\'nski","submitted_at":"2025-07-15T20:48:30Z","abstract_excerpt":"We provide new high-accuracy randomized algorithms for solving linear systems and regression problems that are well-conditioned except for $k$ large singular values. For solving such $d \\times d$ positive definite system our algorithms succeed whp. and run in time $\\tilde O(d^2 + k^\\omega)$. For solving such regression problems in a matrix $\\mathbf{A} \\in \\mathbb{R}^{n \\times d}$ our methods succeed whp. and run in time $\\tilde O(\\mathrm{nnz}(\\mathbf{A}) + d^2 + k^\\omega)$ where $\\omega$ is the matrix multiplication exponent and $\\mathrm{nnz}(\\mathbf{A})$ is the number of non-zeros in $\\mathbf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11724","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.11724/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}