K-Tensors: Clustering Positive Semi-Definite Matrices

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.