{"paper":{"title":"Quasi-optimal hierarchically semi-separable matrix approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.NA"],"primary_cat":"math.NA","authors_text":"Cameron Musco, Christopher Musco, David Persson, Diana Halikias, Feyza Duman Keles, Noah Amsel, Tyler Chen","submitted_at":"2025-05-22T17:27:40Z","abstract_excerpt":"We present a randomized algorithm for producing a quasi-optimal hierarchically semi-separable (HSS) approximation to an $N\\times N$ matrix $A$ using only matrix-vector products with $A$ and $A^T$. We prove that, using $O(k \\log(N/k))$ matrix-vector products and ${O}(N k^2 \\log(N/k))$ additional runtime, the algorithm returns an HSS matrix $B$ with rank-$k$ blocks whose expected Frobenius norm error $\\mathbb{E}[\\|A - B\\|_F^2]$ is at most $O(\\log(N/k))$ times worse than the best possible approximation error by an HSS rank-$k$ matrix. In fact, the algorithm we analyze in a simple modification of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.16937","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/2505.16937/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"}