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

Renjie Xu, Tong Wei, Yimin Wei, Pengpeng Xie

TL;DR

The paper develops QR decompositions for dual matrices, including DQR, thin DQR (tDQR), and randomized DQR with column pivoting (RDQRCP), establishing existence, perturbation bounds, and efficient algorithms for large-scale problems. By treating dual matrices as first-order perturbations $A=A_s+A_i\epsilon$, the authors derive explicit expressions for the infinitesimal parts of the factors and prove that the Q-factor satisfies rigorous perturbation bounds, yielding enhanced orthogonality. They use these dual QR factorizations to compute the dual Moore–Penrose generalized inverse (DMPGI) and to perform proper orthogonal decomposition for traveling-wave identification in time-series data, including large-scale fMRI. Numerical experiments compare performance across DQR variants, confirm the perturbation bounds, and demonstrate improved traveling-wave detection in brain networks, with RDQRCP offering favorable scalability in low-rank settings. Overall, the work provides a rigorous, scalable framework for dual-matrix decompositions with practical impact in brain dynamics and signal processing.

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.

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

TL;DR

The paper develops QR decompositions for dual matrices, including DQR, thin DQR (tDQR), and randomized DQR with column pivoting (RDQRCP), establishing existence, perturbation bounds, and efficient algorithms for large-scale problems. By treating dual matrices as first-order perturbations , the authors derive explicit expressions for the infinitesimal parts of the factors and prove that the Q-factor satisfies rigorous perturbation bounds, yielding enhanced orthogonality. They use these dual QR factorizations to compute the dual Moore–Penrose generalized inverse (DMPGI) and to perform proper orthogonal decomposition for traveling-wave identification in time-series data, including large-scale fMRI. Numerical experiments compare performance across DQR variants, confirm the perturbation bounds, and demonstrate improved traveling-wave detection in brain networks, with RDQRCP offering favorable scalability in low-rank settings. Overall, the work provides a rigorous, scalable framework for dual-matrix decompositions with practical impact in brain dynamics and signal processing.

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.
Paper Structure (15 sections, 6 theorems, 68 equations, 3 figures, 4 tables, 6 algorithms)

This paper contains 15 sections, 6 theorems, 68 equations, 3 figures, 4 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual numbers
  4. Sylvester equation
  5. QR Decomposition of Dual Matrices
  6. QR Decomposition of Dual Matrices (DQR)
  7. Thin QR Decomposition of Dual Matrices (tDQR)
  8. Randomized QR Decomposition with Column Pivoting of Dual Matrices (RDQRCP)
  9. Perturbation Analysis of the Q-factor
  10. Numerical Experiments
  11. Numerical Comparison
  12. Dual Moore-Penrose Generalized Inverse
  13. Simulations of Standing and Traveling Waves
  14. Traveling Wave Identification in the Brain
  15. Proof of Theorem 3.2 and Theorem 3.3

Key Result

Proposition 2.1

qi2021eigenvalues Suppose that $A = (a_{ij}) = A_{s}+A_{i} \epsilon \in \mathbb{DR}^{m\times n}$, $B = B_{s} +B_{i}\epsilon \in \mathbb{DR}^{n\times p}$ and $C =C_{s} +C_{i}\epsilon \in \mathbb{DR}^{n \times n}$. Then

Figures (3)

  • Figure 1: The simulations of pure standing waves exhibit the center positions in the standard part of $Q$ and small values in the infinitesimal part. Simulations of pure traveling waves reveal paired similarities between the standard and infinitesimal parts of $Q$. Leveraging the QR decomposition of the dual matrix and the specific properties of traveling waves, we accurately identify traveling waves within a combination wave composed of four standing waves and two traveling waves.
  • Figure 2: Inner product between vectors of the standard part and infinitesimal part in the CDSVD and the RDQRCP decomposition
  • Figure 3: Paths of traveling waves including Wernicke’s stream and Broca’s stream.

Theorems & Definitions (9)

  • Proposition 2.1
  • Definition 2.1
  • Theorem 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 5.1
  • Theorem 5.1
  • Remark 5.1