{"paper":{"title":"Coordinatizing Data With Lens Spaces and Persistent Cohomology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.AT","authors_text":"Jose A. Perea, Luis Polanco","submitted_at":"2019-05-01T15:47:18Z","abstract_excerpt":"We introduce here a framework to construct coordinates in \\emph{finite} Lens spaces for data with nontrivial 1-dimensional $\\mathbb{Z}_q$ persistent cohomology, $q\\geq 3$. Said coordinates are defined on an open neighborhood of the data, yet constructed with only a small subset of landmarks. We also introduce a dimensionality reduction scheme in $S^{2n-1}/\\mathbb{Z}_q$ (Lens-PCA: $\\mathsf{LPCA}$), and demonstrate the efficacy of the pipeline $PH^1(\\;\\cdot\\; ; \\mathbb{Z}_q)$ class $\\Rightarrow$ $S^{2n-1}/\\mathbb{Z}_q$ coordinates $\\Rightarrow$ $\\mathsf{LPCA}$, for nonlinear (topological) dimens"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.00350","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":""},"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"}