{"paper":{"title":"Hashing embeddings of optimal dimension, with applications to linear least squares","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.NA","math.OC","stat.ML"],"primary_cat":"math.NA","authors_text":"Coralia Cartis, Jan Fiala, Zhen Shao","submitted_at":"2021-05-25T10:35:13Z","abstract_excerpt":"The aim of this paper is two-fold: firstly, to present subspace embedding properties for $s$-hashing sketching matrices, with $s\\geq 1$, that are optimal in the projection dimension $m$ of the sketch, namely, $m=\\mathcal{O}(d)$, where $d$ is the dimension of the subspace. A diverse set of results are presented that address the case when the input matrix has sufficiently low coherence (thus removing the $\\log^2 d$ factor dependence in $m$, in the low-coherence result of Bourgain et al (2015) at the expense of a smaller coherence requirement); how this coherence changes with the number $s$ of co"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.11815","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/2105.11815/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"}