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

TL;DR

This paper introduces a new metric for dual quaternion matrices, reformulates related equations as least squares problems, and proposes algorithms with proven convergence, validated by simulations on synthetic and image data.

Contribution

It presents a novel metric function for dual quaternion matrices and develops proximal point algorithms with convergence analysis for solving related equations.

Findings

Algorithms effectively solve dual quaternion overdetermined equations

Convergence theorems established for the proposed methods

Simulation results demonstrate algorithm effectiveness

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.