QR Decomposition of Dual Matrices and its Application to Traveling Wave Identification in the Brain

TL;DR

This paper introduces a QR decomposition algorithm for dual number matrices and demonstrates its application in improving traveling wave identification in brain fMRI data, enhancing precision and applicability in large-scale problems.

Contribution

It presents a novel QR decomposition method for dual matrices, including explicit solutions and algorithms for large-scale problems, with applications in brain wave analysis.

Findings

Enhanced orthogonality and perturbation bounds for dual matrix QR decomposition

Improved accuracy in computing DMPGI using dual matrix QR decomposition

Significant improvement in identifying brain wave signals in fMRI data

Abstract

Matrix decompositions in dual number representations have played an important role in fields such as kinematics and computer graphics in recent years. In this paper, we present a QR decomposition algorithm for dual number matrices, specifically geared towards its application in traveling wave identification, utilizing the concept of proper orthogonal decomposition. When dealing with large-scale problems, we present explicit solutions for the QR, thin QR, and randomized QR decompositions of dual number matrices, along with their respective algorithms with column pivoting. The QR decomposition of dual matrices is an accurate first-order perturbation, with the Q-factor satisfying rigorous perturbation bounds, leading to enhanced orthogonality. In numerical experiments, we discuss the suitability of different QR algorithms when confronted with various large-scale dual matrices, providing their respective domains of applicability. Subsequently, we employed the QR decomposition of dual matrices to compute the DMPGI, thereby attaining results of higher precision. Moreover, we apply the QR decomposition in the context of traveling wave identification, employing the notion of proper orthogonal decomposition to perform a validation analysis of large-scale functional magnetic resonance imaging (fMRI) data for brain functional circuits. Our approach significantly improves the identification of two types of wave signals compared to previous research, providing empirical evidence for cognitive neuroscience theories.