Fast $(1+\epsilon)$-approximation of the L\"owner extremal matrices of high-dimensional symmetric matrices

TL;DR

This paper introduces a fast, iterative algorithm for approximating extremal matrices in high-dimensional symmetric matrix sets, with applications in biomedical imaging and matrix clustering.

Contribution

It presents a novel, efficient method for approximating L"owner extremal matrices, improving computational speed and simplicity over existing approaches.

Findings

Algorithm achieves high-accuracy approximations quickly

Applicable to large-scale high-dimensional matrix data

Extensions to matrix clustering are discussed

Abstract

Matrix data sets are common nowadays like in biomedical imaging where the Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) modality produces data sets of 3D symmetric positive definite matrices anchored at voxel positions capturing the anisotropic diffusion properties of water molecules in biological tissues. The space of symmetric matrices can be partially ordered using the L\"owner ordering, and computing extremal matrices dominating a given set of matrices is a basic primitive used in matrix-valued signal processing. In this letter, we design a fast and easy-to-implement iterative algorithm to approximate arbitrarily finely these extremal matrices. Finally, we discuss on extensions to matrix clustering.