{"paper":{"title":"Minimax Lower Bounds for Linear Independence Testing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","cs.LG","math.IT","math.ST","stat.TH"],"primary_cat":"stat.ML","authors_text":"Aaditya Ramdas, Aarti Singh, David Isenberg, Larry Wasserman","submitted_at":"2016-01-23T10:20:58Z","abstract_excerpt":"Linear independence testing is a fundamental information-theoretic and statistical problem that can be posed as follows: given $n$ points $\\{(X_i,Y_i)\\}^n_{i=1}$ from a $p+q$ dimensional multivariate distribution where $X_i \\in \\mathbb{R}^p$ and $Y_i \\in\\mathbb{R}^q$, determine whether $a^T X$ and $b^T Y$ are uncorrelated for every $a \\in \\mathbb{R}^p, b\\in \\mathbb{R}^q$ or not. We give minimax lower bound for this problem (when $p+q,n \\to \\infty$, $(p+q)/n \\leq \\kappa < \\infty$, without sparsity assumptions). In summary, our results imply that $n$ must be at least as large as $\\sqrt {pq}/\\|\\S"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.06259","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"}