{"paper":{"title":"An Online Sparsification Algorithm from the Book","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Gernot Z\\\"ocklein, Gramoz Goranci, Maximilian Probst Gutenberg, Rasmus Kyng, Yibin Zhao","submitted_at":"2026-07-25T08:17:55Z","abstract_excerpt":"In their seminal paper [Cohen et al., 2016], Cohen, Musco, and Pachocki proposed a natural and simple online spectral sparsification algorithm: rows $a_1, a_2, \\ldots \\in \\mathbb{R}^d$ of a matrix $A$ arrive one-by-one, and when row $a_i$ arrives, it is appended to sparsifier $\\tilde{A}$ (after appropriately reweighting it) with probability proportional to its current leverage score $$ \\tau^{\\mathrm{OL}}(a_i)=a_i^\\top(A_i^\\top A_i)^\\dagger a_i, \\text{ where }A_i = [a_1, a_2, \\ldots, a_i]^\\top $$ or otherwise discarded forever. For oblivious streams, they showed that this maintains a $(1\\pm\\eps"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23098","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/2607.23098/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"}