Covariance data as represented by symmetric positive definite (SPD) matrices are ubiquitous throughout technical study as efficient descriptors of interdependent systems. Euclidean analysis of SPD matrices, while computationally fast, can lead to skewed and even unphysical interpretations of data. Riemannian methods preserve the geometric structure of SPD data at the cost of expensive eigenvalue computations. In this paper, we propose a geometric method for unsupervised clustering of SPD data based on the Thompson metric. This technique relies upon a novel "inductive midrange" centroid computation for SPD data, whose properties are examined and numerically confirmed. We demonstrate the incorporation of the Thompson metric and inductive midrange into X-means and K-means++ clustering algorithms.

翻译：以正对数确定矩阵(SPD)为代表的共变数据,在整个技术研究中,作为相互依存系统的有效描述符,无处不在。对SPD矩阵的欧几里德分析,虽然计算速度快,但可能导致数据被扭曲甚至非物理解释。里曼方法维护SPD数据的几何结构,以昂贵的损耗值计算为代价。在本文中,我们根据Thompson衡量标准,为SPD数据不受监督的组合提出了一种几何方法。这一技术依靠对SPD数据进行新型的“内导中程”机器人计算,这些数据的特性经过审查和数字确认。我们展示了将Thompson计量和中导中程纳入X值和K means+组算法的情况。