Geometric Learning and Finsler Metrics in Weighted Projective Spaces

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.