{"paper":{"title":"Additive Approximation Schemes for Low-Dimensional Embeddings","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Ainesh Bakshi, Prashanti Anderson, Samuel B. Hopkins","submitted_at":"2025-09-11T17:45:21Z","abstract_excerpt":"We consider the task of fitting low-dimensional embeddings to high-dimensional data. In particular, we study the $k$-Euclidean Metric Violation problem ($\\textsf{$k$-EMV}$), where the input is $D \\in \\mathbb{R}^{\\binom{n}{2}}_{\\geq 0}$ and the goal is to find the closest vector $X \\in \\mathbb{M}_{k}$, where $\\mathbb{M}_k \\subset \\mathbb{R}^{\\binom{n}{2}}_{\\geq 0}$ is the set of all $k$-dimensional Euclidean metrics on $n$ points, and closeness is formulated as the following optimization problem, where $\\| \\cdot \\|$ is the entry-wise $\\ell_2$ norm: \\[\n  \\textsf{OPT}_{\\textrm{EMV}} = \\min_{X \\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.09652","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.09652/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}