This paper introduces a novel self-consistency clustering algorithm (K-Tensors) designed for positive-semidefinite matrices based on their eigenstructures. As positive semi-definite matrices can be represented as ellipses or ellipsoids in $\Re^p$, $p \ge 2$, it is critical to maintain their structural information to perform effective clustering. However, traditional clustering algorithms often vectorize the matrices, resulting in a loss of essential structural information. To address this issue, we propose a distance metric that is specifically based on the structural information of positive semi-definite matrices. This distance metric enables the clustering algorithm to consider the differences between positive semi-definite matrices and their projection onto the common space spanned by a set of positive semi-definite matrices. This innovative approach to clustering positive semi-definite matrices has broad applications in several domains, including financial and biomedical research, such as analyzing functional connectivity data. By maintaining the structural information of positive semi-definite matrices, our proposed algorithm promises to cluster the positive semi-definite matrices in a more meaningful way, thereby facilitating deeper insights into the underlying data in various applications.

翻译：本文提出了一种新颖的自一致性聚类算法（K-Tensors），该算法基于特征结构针对半正定矩阵设计。由于半正定矩阵可表示为$\Re^p$（$p \ge 2$）中的椭圆或椭球体，保留其结构信息对于实现有效聚类至关重要。然而，传统聚类算法通常将矩阵向量化，导致关键结构信息的丢失。为解决这一问题，我们提出了一种专门基于半正定矩阵结构信息的距离度量。该距离度量使聚类算法能够考虑半正定矩阵之间的差异，以及它们到一组半正定矩阵所张成的公共空间的投影。这种半正定矩阵的创新聚类方法在金融和生物医学研究等多个领域具有广泛应用，例如分析功能连接数据。通过保留半正定矩阵的结构信息，所提出的算法有望以更有意义的方式对半正定矩阵进行聚类，从而促进对各类应用中底层数据更深入的洞察。