[FORTHCOMING] Geometric learning and Finsler metrics in weighted projective spaces

We introduce a hierarchical clustering framework for weighted projective spaces $$$ℙ_{\parallel}$$$ 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 $$$d_{F,ℚ}([z], [w])$$$ for points of $$$ℙ_{\parallel}(ℚ)$$$. 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 $$$ℙ_{\parallel}$$$, 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 $$${\parallel}$$$ 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.

We introduce a hierarchical clustering framework for weighted projective spaces $$$ℙ_{\parallel}$$$ 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 $$$d_{F,ℚ}([z], [w])$$$ for points of $$$ℙ_{\parallel}(ℚ)$$$. 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 $$$ℙ_{\parallel}$$$, 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 $$${\parallel}$$$ 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.

Geometric clustering Weighted projective spaces Finsler metrics Non-Euclidean manifolds

Geometric clustering Weighted projective spaces Finsler metrics Non-Euclidean manifolds