{"paper":{"title":"Geometric Learning and Finsler Metrics in Weighted Projective Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.DG","authors_text":"Tony Shaska","submitted_at":"2025-05-07T21:57:27Z","abstract_excerpt":"We introduce a hierarchical clustering framework for weighted projective spaces $\\mathbb{P}_{\\mathbf{q}}$ built on Finsler geometry. From an optimization-based Finsler norm that quotients out the weighted scaling action, we construct a scaling-invariant distance $d_F([z], [w])$ and a rational analogue $d_{F,\\mathbb{Q}}([z], [w])$ for points of $\\mathbb{P}_{\\mathbf{q}}(\\mathbb{Q})$. The norm carries a shape parameter $p$: the case $p=2$ is Riemannian and admits a closed-form distance, while $p\\neq 2$ is genuinely Finsler, and the metric and clustering guarantees below hold for every $p\\in[1,\\in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.00001","kind":"arxiv","version":3},"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/2507.00001/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"}