{"paper":{"title":"Computational Hardness of Certifying Bounds on Constrained PCA Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.ST","stat.TH"],"primary_cat":"cs.DS","authors_text":"Afonso S. Bandeira, Alexander S. Wein, Dmitriy Kunisky","submitted_at":"2019-02-19T22:18:46Z","abstract_excerpt":"Given a random $n \\times n$ symmetric matrix $\\boldsymbol W$ drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form $\\boldsymbol x^\\top \\boldsymbol W \\boldsymbol x$ over all vectors $\\boldsymbol x$ in a constraint set $\\mathcal{S} \\subset \\mathbb{R}^n$. For a certain class of normalized constraint sets $\\mathcal{S}$ we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of $\\boldsymbol W$. A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1902.07324","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"}