{"paper":{"title":"Perturbation of linear forms of singular vectors under Gaussian noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dong Xia, Vladimir Koltchinskii","submitted_at":"2015-06-09T03:26:27Z","abstract_excerpt":"Let $A\\in\\mathbb{R}^{m\\times n}$ be a matrix of rank $r$ with singular value decomposition (SVD) $A=\\sum_{k=1}^r\\sigma_k (u_k\\otimes v_k),$ where $\\{\\sigma_k, k=1,\\ldots,r\\}$ are singular values of $A$ (arranged in a non-increasing order) and $u_k\\in {\\mathbb R}^m, v_k\\in {\\mathbb R}^n, k=1,\\ldots, r$ are the corresponding left and right orthonormal singular vectors. Let $\\tilde{A}=A+X$ be a noisy observation of $A,$ where $X\\in\\mathbb{R}^{m\\times n}$ is a random matrix with i.i.d. Gaussian entries, $X_{ij}\\sim\\mathcal{N}(0,\\tau^2),$ and consider its SVD $\\tilde{A}=\\sum_{k=1}^{m\\wedge n}\\tilde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02764","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"}