# K-Tensors: Clustering Positive Semi-Definite Matrices

arXiv: https://arxiv.org/abs/2306.06534 · HTML: https://mlsanity.com/paper/2306.06534/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 41 references that are in the map.
- First posted on arXiv 10 Jun 2023.

## Builds on (most cited first, 7 of 7)

- [Statistical Model for Dynamically-Changing Correlation Matrices with Application to Brain Connectivity](https://mlsanity.com/paper/1812.10050/) (2018)
- [Random Projections on Manifolds of Symmetric Positive Definite Matrices for Image Classification](https://mlsanity.com/paper/1403.0700/) (2014)
- [Shared Subspace Models for Multi-Group Covariance Estimation](https://mlsanity.com/paper/1607.03045/) (2016)
- [Principal Regression for High Dimensional Covariance Matrices](https://mlsanity.com/paper/2007.12740/) (2020)
- [k-means on Positive Definite Matrices, and an Application to Clustering in Radar Image Sequences](https://mlsanity.com/paper/2008.03454/) (2020)
- [Geometric Machine Learning for Channel Covariance Estimation in Vehicular Networks](https://mlsanity.com/paper/2107.00759/) (2021)
- [Longitudinal regression of covariance matrix outcomes](https://mlsanity.com/paper/2202.04553/) (2022)

## Abstract (arXiv)

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.