This paper introduces a novel self-consistency clustering algorithm ($K$-Tensors) designed for {partitioning a distribution of} positive-semidefinite matrices based on their eigenstructures. As positive semi-definite matrices can be represented as ellipsoids in $\Re^p$, $p \ge 2$, it is critical to maintain their structural information to perform effective clustering. However, traditional clustering algorithms {applied to matrices} often {involve vectorization of} the matrices, resulting in a loss of essential structural information. To address this issue, we propose a distance metric {for clustering} 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 projections onto {a} common space spanned by \thadJulyTen{orthonormal vectors defined from 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聚类算法（$K$-Tensors），旨在基于特征结构对正半定矩阵分布进行划分。由于正半定矩阵可表示为 $\Re^p$（$p \ge 2$）空间中的椭球体，保留其结构信息对于实现有效聚类至关重要。然而，传统聚类算法在处理矩阵时通常涉及向量化操作，导致关键结构信息丢失。为解决这一问题，我们提出了一种专门基于正半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量正半定矩阵与其投影到由一组正半定矩阵定义的规范正交向量张成的公共空间上的差异。这种针对正半定矩阵的创新性聚类方法在金融与生物医学研究（如功能连接性数据分析）等多个领域具有广泛的应用前景。通过保留正半定矩阵的结构信息，我们的算法有望以更具意义的方式实现聚类，从而促进对各类应用场景中潜在数据的深入洞察。