标题： 加权射影空间中的Finsler度量聚类 显示英文标题

标题： Finsler Metric Clustering in Weighted Projective Spaces

摘要： 本文针对加权射影空间 $\mathbb{P}_{\mathbf{q}}$，开发了一种层次聚类算法。该算法利用芬斯勒度量 $ d_F([z], [w]) $ 及其有理类似 $ d_{F,\mathbb{Q}}([z], [w])$ 来定义保持这些商流形非欧几里得几何形状的距离。 这些度量通过对尺度不变的芬斯勒范数进行测地线积分来定义，该积分由等级 $\mathbf{q} = (q_0, q_1, \dots, q_n)$ 加权。这些度量满足包括三角不等式在内的真实度量性质，克服了先前工作中非度量相异性度量的局限性。 显示英文摘要

摘要： This paper develops a hierarchical clustering algorithm for weighted projective spaces $\mathbb{P}_{\mathbf{q}}$, utilizing a Finsler metric $ d_F([z], [w]) $ and its rational analogue $ d_{F,\mathbb{Q}}([z], [w])$ to define distances that preserve the non-Euclidean geometry of these quotient manifolds. Defined via geodesic integrals of a scaling invariant Finsler norm weighted by the grades $\mathbf{q} = (q_0, q_1, \dots, q_n)$, these metrics satisfy true metric properties including the triangle inequality, overcoming the limitations of the non-metric dissimilarity measure from prior work.