{"paper":{"title":"Efficient $\\widetilde{O}(n/\\epsilon)$ Spectral Sketches for the Laplacian and its Pseudoinverse","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"cs.DS","authors_text":"Aaron Sidford, Arun Jambulapati","submitted_at":"2017-11-02T00:06:55Z","abstract_excerpt":"In this paper we consider the problem of efficiently computing $\\epsilon$-sketches for the Laplacian and its pseudoinverse. Given a Laplacian and an error tolerance $\\epsilon$, we seek to construct a function $f$ such that for any vector $x$ (chosen obliviously from $f$), with high probability $(1-\\epsilon) x^\\top A x \\leq f(x) \\leq (1 + \\epsilon) x^\\top A x$ where $A$ is either the Laplacian or its pseudoinverse. Our goal is to construct such a sketch $f$ efficiently and to store it in the least space possible.\n  We provide nearly-linear time algorithms that, when given a Laplacian matrix $\\m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.00571","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"}