{"paper":{"title":"Randomized biorthogonalization through a two-sided Gram-Schmidt process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Igor Simunec, Laura Grigori, Lorenzo Piccinini","submitted_at":"2025-09-04T16:49:50Z","abstract_excerpt":"We propose and analyze a randomized two-sided Gram-Schmidt process for the biorthogonalization of two given matrices $X, Y \\in\\mathbb{R}^{n\\times m}$. The algorithm aims to find two matrices $Q, P \\in\\mathbb{R}^{n\\times m}$ such that ${\\rm range}(X) = {\\rm range}(Q)$, ${\\rm range}(Y) = {\\rm range}(P)$ and $(\\Omega Q)^T \\Omega P = I$, where $\\Omega \\in\\mathbb{R}^{s \\times n}$ is a sketching matrix satisfying an oblivious subspace $\\varepsilon$-embedding property; in other words, the biorthogonality condition on the columns of $Q$ and $P$ is replaced by an equivalent condition on their sketches."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.04386","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/2509.04386/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"}