{"paper":{"title":"On the Gap Between Strict-Saddles and True Convexity: An Omega(log d) Lower Bound for Eigenvector Approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.IT","math.CO","math.IT","stat.ML"],"primary_cat":"cs.LG","authors_text":"Ahmed El Alaoui, Benjamin Recht, Max Simchowitz","submitted_at":"2017-04-14T21:56:11Z","abstract_excerpt":"We prove a \\emph{query complexity} lower bound on rank-one principal component analysis (PCA). We consider an oracle model where, given a symmetric matrix $M \\in \\mathbb{R}^{d \\times d}$, an algorithm is allowed to make $T$ \\emph{exact} queries of the form $w^{(i)} = Mv^{(i)}$ for $i \\in \\{1,\\dots,T\\}$, where $v^{(i)}$ is drawn from a distribution which depends arbitrarily on the past queries and measurements $\\{v^{(j)},w^{(j)}\\}_{1 \\le j \\le i-1}$. We show that for a small constant $\\epsilon$, any adaptive, randomized algorithm which can find a unit vector $\\widehat{v}$ for which $\\widehat{v}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.04548","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"}