{"paper":{"title":"Metricizing the Euclidean Space towards Desired Distance Relations in Point Clouds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.LG"],"primary_cat":"cs.CG","authors_text":"Maksim Goman, Sandra K\\\"onig, Shahzad Ahmad, Stefan Rass","submitted_at":"2022-11-07T16:37:29Z","abstract_excerpt":"Given a set of points in the Euclidean space $\\mathbb{R}^\\ell$ with $\\ell>1$, the pairwise distances between the points are determined by their spatial location and the metric $d$ that we endow $\\mathbb{R}^\\ell$ with. Hence, the distance $d(\\mathbf x,\\mathbf y)=\\delta$ between two points is fixed by the choice of $\\mathbf x$ and $\\mathbf y$ and $d$. We study the related problem of fixing the value $\\delta$, and the points $\\mathbf x,\\mathbf y$, and ask if there is a topological metric $d$ that computes the desired distance $\\delta$. We demonstrate this problem to be solvable by constructing a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.03674","kind":"arxiv","version":2},"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/2211.03674/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"}