Geometric Learning and Finsler Metrics in Weighted Projective Spaces

Mathematics > Differential Geometry [Submitted on 7 May 2025 (v1), last revised 15 Jun 2026 (this version, v3)] Title:Geometric Learning and Finsler Metrics in Weighted Projective Spaces View PDF HTML (experimental)Abstract: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 $p\in[1,\infty)$. Whereas earlier work measured proximity in these spaces through non-metric dissimilarities, we prove that $d_F$ satisfies the triangle inequality and is therefore a genuine metric; this is what equips the induced clustering with its theoretical guarantees, including monotone dendrograms and Gromov--Hausdorff stability under perturbation of the data. The metric respects the intrinsic scaling symmetry and weighted topology of $\mathbb{P}_{\mathbf{q}}$, avoiding the distortions of a flat-space embedding. We develop the framework's arithmetic applications -- clustering rational points in the moduli space of genus two curves and analyzing rational functions in arithmetic dynamics -- and indicate prospective extensions to quantum state spaces, where the weights $\mathbf{q}$ model anisotropic noise. More broadly, the construction offers a rigorous metric foundation for graded neural networks and related machine-learning techniques on graded algebraic varieties. Submission history From: Tony Shaska Sr [view email][v1] Wed, 7 May 2025 21:57:27 UTC (23 KB) [v2] Mon, 5 Jan 2026 16:51:32 UTC (25 KB) [v3] Mon, 15 Jun 2026 21:38:47 UTC (30 KB) Current browse context: math.DG References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.