This paper introduces $K$-Tensors, a novel self-consistent clustering algorithm designed to cluster positive semi-definite (PSD) matrices by their eigenstructures. Clustering PSD matrices is crucial across various fields, including computer and biomedical sciences. Traditional clustering methods, which often involve matrix vectorization, tend to overlook the inherent PSD characteristics, thereby discarding valuable shape and eigenstructural information. To preserve this essential shape and eigenstructral information, our approach incorporates a unique distance metric that respects the PSD nature of the data. We demonstrate that $K$-Tensors is not only self-consistent but also reliably converges to a local optimum. Through numerical studies, we further validate the algorithm's effectiveness and explore its properties in detail.

翻译：本文提出了$K$-张量（$K$-Tensors）这一新颖的自一致性聚类算法，旨在根据特征结构对半正定矩阵进行聚类。聚类半正定矩阵在计算机科学和生物医学等多个领域具有关键意义。传统聚类方法常涉及矩阵向量化，倾向于忽略半正定矩阵固有的特性，从而丢弃了有价值的形状和特征结构信息。为保留这些关键的形状与特征结构信息，我们的方法引入了一种尊重数据半正定特性的独特距离度量。我们证明了$K$-张量不仅具有自一致性，而且能可靠地收敛至局部最优解。通过数值研究，我们进一步验证了该算法的有效性，并详细探究了其性质。