{"paper":{"title":"Near-optimal hierarchical matrix approximation from matrix-vector products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA","math.NA"],"primary_cat":"cs.DS","authors_text":"Cameron Musco, Christopher Musco, David Persson, Diana Halikias, Feyza Duman Keles, Tyler Chen","submitted_at":"2024-07-05T17:49:15Z","abstract_excerpt":"We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\\times n$ matrix $\\mathbf{A}$, accessible only though matrix-vector products with $\\mathbf{A}$ and $\\mathbf{A}^{\\mathsf{T}}$. We prove that, for the rank-$k$ HODLR approximation problem, our method achieves a $(1+\\beta)^{\\log(n)}$-optimal approximation in expected Frobenius norm using $O(k\\log(n)/\\beta^3)$ matrix-vector products. In particular, the algorithm obtains a $(1+\\varepsilon)$-optimal approximation with $O(k\\log^4(n)/\\varepsilon^3)$ matrix-vector products, a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.04686","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2407.04686/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"}