{"paper":{"title":"Low-distortion Subspace Embeddings in Input-sparsity Time and Applications to Robust Linear Regression","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Michael W. Mahoney, Xiangrui Meng","submitted_at":"2012-10-11T06:18:59Z","abstract_excerpt":"Low-distortion embeddings are critical building blocks for developing random sampling and random projection algorithms for linear algebra problems. We show that, given a matrix $A \\in \\R^{n \\times d}$ with $n \\gg d$ and a $p \\in [1, 2)$, with a constant probability, we can construct a low-distortion embedding matrix $\\Pi \\in \\R^{O(\\poly(d)) \\times n}$ that embeds $\\A_p$, the $\\ell_p$ subspace spanned by $A$'s columns, into $(\\R^{O(\\poly(d))}, \\| \\cdot \\|_p)$; the distortion of our embeddings is only $O(\\poly(d))$, and we can compute $\\Pi A$ in $O(\\nnz(A))$ time, i.e., input-sparsity time. Our "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.3135","kind":"arxiv","version":3},"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"}