{"paper":{"title":"Approximating Sparsest Cut in Low Rank Graphs via Embeddings from Approximately Low-Dimensional Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Rakesh Venkat, Yuval Rabani","submitted_at":"2017-06-21T09:38:37Z","abstract_excerpt":"We consider the problem of embedding a finite set of points $\\{x_1, \\ldots, x_n\\} \\in \\mathbb{R}^d$ that satisfy $\\ell_2^2$ triangle inequalities into $\\ell_1$, when the points are approximately low-dimensional. Goemans (unpublished, appears in a work of [Magen and Moharammi, 2008]) showed that such points residing in \\emph{exactly} $d$ dimensions can be embedded into $\\ell_1$ with distortion at most $\\sqrt{d}$. We prove the following robust analogue of this statement: if there exists a $r$-dimensional subspace $\\Pi$ such that the projections onto this subspace satisfy $\\sum_{i,j \\in [n]}\\Vert"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06806","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"}