{"paper":{"title":"A Probabilistic Subspace Bound with Application to Active Subspaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Ilse C.F. Ipsen, John T. Holodnak, Ralph C. Smith","submitted_at":"2018-01-02T15:19:20Z","abstract_excerpt":"Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions but only on the numerical rank (intrinsic dimension) of E. Our approach resembles the low-rank approximation of kernel matrices from random features, but our accuracy measures are more stringent.\n  In the context of parameter selection based on active subspaces, where S is computed via Monte Carlo sampling, we present a bound on the number"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.00682","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":""},"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"}