🤖 AI Summary
Existing SPD matrix clustering methods typically rely on vectorization, which disregards the intrinsic Riemannian geometry of the positive semidefinite (PSD) manifold and distorts spectral structure. To address this, we propose K-Tensors—a novel clustering algorithm that operates directly on the PSD manifold without vectorization, thereby preserving geometric and spectral properties. Our key contributions are: (1) a self-consistent clustering framework that maintains eigenstructure fidelity; (2) a new distance metric explicitly designed to respect the geodesic geometry of the PSD manifold; and (3) rigorous theoretical guarantees of convergence to a local optimum. Extensive experiments on multiple benchmark datasets demonstrate that K-Tensors significantly outperforms conventional vectorization-based approaches in clustering accuracy, while robustly retaining matrix shape characteristics and spectral signatures. Moreover, the algorithm exhibits stable convergence behavior and offers strong interpretability through its geometrically grounded formulation.
📝 Abstract
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.