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A metric function for dual quaternion matrices and related least-squares problems

Chen Ling, Chenjian Pan, Liqun Qi

Abstract

Solving dual quaternion equations is an important issue in many fields such as scientific computing and engineering applications. In this paper, we first introduce a new metric function for dual quaternion matrices. Then, we reformulate dual quaternion overdetermined equations as a least squares problem, which is further converted into a bi-level optimization problem. Numerically, we propose two implementable proximal point algorithms for finding approximate solutions of dual quaternion overdetermined equations. The relevant convergence theorems %and computational complexity estimates have also been established. Preliminary simulation results on synthetic and color image datasets demonstrate the effectiveness of the proposed algorithms.

A metric function for dual quaternion matrices and related least-squares problems

Abstract

Solving dual quaternion equations is an important issue in many fields such as scientific computing and engineering applications. In this paper, we first introduce a new metric function for dual quaternion matrices. Then, we reformulate dual quaternion overdetermined equations as a least squares problem, which is further converted into a bi-level optimization problem. Numerically, we propose two implementable proximal point algorithms for finding approximate solutions of dual quaternion overdetermined equations. The relevant convergence theorems %and computational complexity estimates have also been established. Preliminary simulation results on synthetic and color image datasets demonstrate the effectiveness of the proposed algorithms.

Paper Structure

This paper contains 16 sections, 14 theorems, 61 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dual numbers
  4. Quaternion, quaternion matrices and quaternion matrix functions
  5. Dual quaternion and dual quaternion matrices
  6. A dual-metric function on $\dot{\mathbb{Q}}^{m\times n}$
  7. Least-squares problems of dual quaternion equation in the sense of metric function ${\bf\rho}$
  8. Bi-level program formulation
  9. Description of algorithms
  10. Convergence analysis
  11. Convergence of Algorithm \ref{['PrOalg-1']}
  12. Convergence of Algorithm \ref{['alg-2']}
  13. Numerical experiments
  14. Synthetic data
  15. Color images
  16. ...and 1 more sections

Key Result

proposition 1

Let $A\in \mathbb{Q}^{m\times p}$ and $B\in \mathbb{Q}^{p\times n}$. If $m\geq p$, then it holds that $\sigma_{\rm min}(A)\|B\|_F\leq \|AB\|_F$, where $\sigma_{\rm min}(A)$ is the smallest singular value of $A$.

Figures (6)

  • Figure 1: Results of singular values obtained by Alg1 and Alg2 in the synthetic scenario
  • Figure 2: Four test images. From left to right: sailboat, house, peppers and baboon
  • Figure 3: Visual results of $B_{\rm st}$ and $B_{\rm in}$ with "sailboat + house" (first two) and "peppers + baboon" (last two) in case (ii) and $m=300$.
  • Figure 4: Visual results of $X_{\rm st}$ and $X_{\rm in}$ in "sailboat + house". First row: case (i) with $m=180$; Second row: case (ii) with $m=300$; Third row: case (ii) with $m=400$. The first two columns are $X_{\rm st}^\diamond$ obtained by Alg1 and Alg2, and the last two columns are $X_{\rm in}^\diamond$ obtained by Alg1 and Alg2, respectively
  • Figure 5: Visual results of $X_{\rm st}$ and $X_{\rm in}$ in "peppers + baboon". First row: case (i) with $m=180$; Second row: case (ii) with $m=300$; Third row: case (ii) with $m=400$. The first two columns are $X_{\rm st}^\diamond$ obtained by Alg1 and Alg2, and the last two columns are $X_{\rm in}^\diamond$ obtained by Alg1 and Alg2, respectively
  • ...and 1 more figures

Theorems & Definitions (27)

  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • theorem 1
  • proposition 4
  • proposition 5
  • proof
  • ...and 17 more