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arXiv · 2306.06534

K-Tensors: Clustering Positive Semi-Definite Matrices

Hanchao Zhang·Xiaomeng Ju·Baoyi Shi·Lingsong Meng·Thaddeus Tarpey
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Abstract

This paper presents a new clustering algorithm for symmetric positive semi-definite (SPSD) matrices, called K-Tensors. The method identifies structured subsets of the SPSD cone characterized by common principal component (CPC) representations, where each subset corresponds to matrices sharing a common eigenstructure. Unlike conventional clustering approaches that rely on vectorization or transformations of SPSD matrices, thereby losing critical geometric and spectral information, K-Tensors introduces a divergence that respects the intrinsic geometry of SPSD matrices. This divergence preserves the shape and eigenstructure information and yields principal SPSD tensors, defined as a set of representative matrices that summarize the distribution of SPSD matrices. By exploring its theoretical properties, we show that the proposed clustering algorithm is self-consistent under mild distribution assumptions and converges to a local optimum. We demonstrate the use of the algorithm through an application to resting-state functional magnetic resonance imaging (rs-fMRI) data from the Human Connectome Project, where we cluster brain connectivity matrices to discover groups of subjects with shared connectivity structures.

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Hanchao Zhang, Xiaomeng Ju, Baoyi Shi, Lingsong Meng, Thaddeus Tarpey. 2023-06-10. K-Tensors: Clustering Positive Semi-Definite Matrices. https://arxiv.org/abs/2306.06534

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