{"paper":{"title":"Embedding approximately low-dimensional $\\ell_2^2$ metrics into $\\ell_1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Amit Deshpande, Prahladh Harsha, Rakesh Venkat","submitted_at":"2015-12-14T04:54:08Z","abstract_excerpt":"Goemans showed that any $n$ points $x_1, \\dotsc x_n$ in $d$-dimensions satisfying $\\ell_2^2$ triangle inequalities can be embedded into $\\ell_{1}$, with worst-case distortion at most $\\sqrt{d}$. We extend this to the case when the points are approximately low-dimensional, albeit with average distortion guarantees. More precisely, we give an $\\ell_{2}^{2}$-to-$\\ell_{1}$ embedding with average distortion at most the stable rank, $\\mathrm{sr}(M)$, of the matrix $M$ consisting of columns $\\{x_i-x_j\\}_{i<j}$. Average distortion embedding suffices for applications such as the Sparsest Cut problem. O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04170","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"}