{"paper":{"title":"Large distortion dimension reduction using random variable","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Alon Dmitriyuk, Yehoram Gordon","submitted_at":"2013-08-13T06:41:37Z","abstract_excerpt":"Consider a random matrix $H:\\mathbb{R}^n\\longrightarrow\\mathbb{R}^m$. Let $D\\geq2$ and let $\\{W_l\\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\\mathbb{R}^n$. We ask what is the probability that for all $1\\leq l\\leq p$ and $x,y\\in W_l$, \\[\n  \\|x-y\\|_2\\leq\\|Hx-Hy\\|_2\\leq D\\|x-y\\|_2. \\] We show that for $m=O\\big(k+\\frac{\\ln{p}}{\\ln{D}}\\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1308.2768","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"}