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Goemans (unpublished, appears in a work of [Magen and Moharammi, 2008]) showed that such points residing in \\emph{exactly} $d$ dimensions can be embedded into $\\ell_1$ with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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1706.06806","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DS","submitted_at":"2017-06-21T09:38:37Z","cross_cats_sorted":[],"title_canon_sha256":"f323241c913c94e891f1f178e918147de96db4fb38de7e18bf5d6d622e10edcf","abstract_canon_sha256":"3325a566bc78f4a1d79d91e4616ea374c4131628809348dc8cb65d76880f8c7c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:56.812435Z","signature_b64":"aixMLIAETrUDxFh6iJ+BZf7aV42t7K2iWpyPnSLBgDwiWiEeMNxcUELXcNxINmLvn6SKDipEmaiIIGt01G+zDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15b64b85e5d9cedf47ae48c658c309010badf51a635ae8c4e8006cedf054e890","last_reissued_at":"2026-05-18T00:41:56.811732Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:56.811732Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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