Positive semidefinite (PSD) matrices are indispensable in many fields of science. A similarity measurement for such matrices is usually an essential ingredient in the mathematical modelling of a scientific problem. This paper proposes a unified framework to construct similarity measurements for PSD matrices. The framework is obtained by exploring the fiber bundle structure of the cone of PSD matrices and generalizing the idea of the point-set distance previously developed for linear subsapces and positive definite (PD) matrices. The framework demonstrates both theoretical advantages and computational convenience: (1) We prove that the similarity measurement constructed by the framework can be recognized either as the cost of a parallel transport or as the length of a quasi-geodesic curve. (2) We extend commonly used divergences for equidimensional PD matrices to the non-equidimensional case. Examples include Kullback-Leibler divergence, Bhattacharyya divergence and R\'enyi divergence. We prove that these extensions enjoy the same consistency property as their counterpart for geodesic distance. (3) We apply our geometric framework to further extend those in (2) to similarity measurements for arbitrary PSD matrices. We also provide simple formulae to compute these similarity measurements in most situations.

翻译：半正定矩阵在众多科学领域中不可或缺。针对此类矩阵的相似性度量通常是科学问题数学建模中的关键要素。本文提出一个统一框架，用于构建半正定矩阵的相似性度量。该框架通过探索半正定矩阵锥的纤维丛结构，并推广先前针对线性子空间和正定矩阵提出的点集距离思想而获得。该框架兼具理论优势与计算便利性：(1) 我们证明由该框架构建的相似性度量既可视为平行输运的成本，也可视为拟测地曲线的长度；(2) 我们将等维正定矩阵的常用散度扩展到非等维情形，示例包括Kullback-Leibler散度、Bhattacharyya散度和Rényi散度。我们证明这些扩展保持了与测地距离对应版本相同的一致性性质；(3) 我们应用该几何框架进一步将(2)中的度量扩展至任意半正定矩阵的相似性度量。同时，我们为大多数情况下这些相似性度量的计算提供了简洁公式。