Finsler Metric Clustering in Weighted Projective Spaces

TL;DR

This paper introduces a Finsler metric-based hierarchical clustering framework for weighted projective spaces, ensuring metric properties and stability, with applications in arithmetic geometry, dynamics, and quantum state analysis.

Contribution

It develops a true metric Finsler geometry approach for clustering in weighted projective spaces, overcoming limitations of previous dissimilarity measures and preserving intrinsic symmetries.

Findings

Proves the Finsler-derived distance satisfies the triangle inequality.

Demonstrates stability of clustering via Gromov-Hausdorff distance.

Shows applicability in arithmetic geometry and quantum state analysis.

Abstract

This paper establishes a foundational framework for geometric learning in weighted projective spaces $\mathbb{P}_{\mathbb{q}}$ by introducing a hierarchical clustering algorithm governed by Finsler geometry. We define a scaling-invariant Finsler metric $d_F([z], [w])$-and its rational analogue $d_{F,\mathbb{Q}}([z], [w])$-derived from an optimization-based Finsler norm that effectively quotients out the weighted scaling action. Unlike previous approaches that characterized these spaces via non-metric dissimilarity measures, we rigorously prove that our construction satisfies the triangle inequality, providing a true metric framework that ensures the stability of hierarchical clustering via the Gromov-Hausdorff distance. We demonstrate that this metric approach preserves the intrinsic scaling symmetries and weighted topology of $\mathbb{P}_{\mathbb{q}}$ without the topological distortions inherent in Euclidean approximations. The algorithm's efficacy is explored in the context of arithmetic geometry (clustering moduli spaces of genus two curves), arithmetic dynamics, and quantum state-space analysis, where the weights $\mathbb{q}$ represent anisotropic physical constraints and noise profiles. This work establishes a robust theoretical foundation for the development of graded neural networks and other machine learning techniques for graded algebraic varieties.