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Improved two-block coordinate descent method for Pose Graph Optimization Problem under $F^*$-norm

Yongjun Chen, Liping Zhang

TL;DR

The paper addresses pose graph optimization (PGO) in SLAM by reframing it as a rank-one dual quaternion Hermitian matrix completion task and reformulating the objective under the $F^*$-norm to better capture dual-part magnitudes. It develops an explicit framework for optimal rank-$k$ approximation under both the $F$-norm and the $F^*$-norm via the dual complex adjoint, and it embeds these results into an improved two-block coordinate descent method for PGO under the $F^*$-norm. It introduces a robust spectral initialization and stagnation-based termination alongside proper parameter tuning to accelerate convergence. Experimental results on synthetic PGO instances show higher accuracy, faster computation, and higher success rates, especially in low-observation regimes, with the $F^*$-norm outperforming the traditional $F$-norm in capturing dual-part magnitudes.

Abstract

Dual quaternions and dual quaternion matrices are widely used in robotics research, particularly in simultaneous localization and mapping (SLAM) problem. Using dual quaternion theory and graph-based methods, SLAM can be reformulated as a rank-one dual quaternion Hermitian matrix completion problem, known as the pose graph optimization (PGO) problem. Recently, Qi and Cui introduced a two-block coordinate descent method to solve this reformulated problem. In this paper, we enhance this method by reformulating the PGO problem under the more appropriate and robust F*-norm rather than the conventional Frobenius norm, leading to improved experimental accuracy. We show that under the F*-norm, one block has a closed-form solution and another is the optimal rank-one approximation of dual quaternion Hermitian matrices under the F*-norm. We derive an explicit solution for this approximation and present an efficient algorithm to compute it. To further enhance the two-block coordinate descent method, we introduce proper parameter selection, stagnation-based termination criteria and an effective spectral initialization strategy. Extensive numerical experiments demonstrate that our refinements deliver superior accuracy, faster computation, and higher success rates, particularly in low-observation settings. In particular, using the F*-norm outperforms the traditional F-norm, underscoring its ability to more faithfully capture the magnitude of the dual parts of dual quaternion matrices.

Improved two-block coordinate descent method for Pose Graph Optimization Problem under $F^*$-norm

TL;DR

The paper addresses pose graph optimization (PGO) in SLAM by reframing it as a rank-one dual quaternion Hermitian matrix completion task and reformulating the objective under the -norm to better capture dual-part magnitudes. It develops an explicit framework for optimal rank- approximation under both the -norm and the -norm via the dual complex adjoint, and it embeds these results into an improved two-block coordinate descent method for PGO under the -norm. It introduces a robust spectral initialization and stagnation-based termination alongside proper parameter tuning to accelerate convergence. Experimental results on synthetic PGO instances show higher accuracy, faster computation, and higher success rates, especially in low-observation regimes, with the -norm outperforming the traditional -norm in capturing dual-part magnitudes.

Abstract

Dual quaternions and dual quaternion matrices are widely used in robotics research, particularly in simultaneous localization and mapping (SLAM) problem. Using dual quaternion theory and graph-based methods, SLAM can be reformulated as a rank-one dual quaternion Hermitian matrix completion problem, known as the pose graph optimization (PGO) problem. Recently, Qi and Cui introduced a two-block coordinate descent method to solve this reformulated problem. In this paper, we enhance this method by reformulating the PGO problem under the more appropriate and robust F*-norm rather than the conventional Frobenius norm, leading to improved experimental accuracy. We show that under the F*-norm, one block has a closed-form solution and another is the optimal rank-one approximation of dual quaternion Hermitian matrices under the F*-norm. We derive an explicit solution for this approximation and present an efficient algorithm to compute it. To further enhance the two-block coordinate descent method, we introduce proper parameter selection, stagnation-based termination criteria and an effective spectral initialization strategy. Extensive numerical experiments demonstrate that our refinements deliver superior accuracy, faster computation, and higher success rates, particularly in low-observation settings. In particular, using the F*-norm outperforms the traditional F-norm, underscoring its ability to more faithfully capture the magnitude of the dual parts of dual quaternion matrices.
Paper Structure (11 sections, 12 theorems, 72 equations, 1 figure, 5 tables, 7 algorithms)

This paper contains 11 sections, 12 theorems, 72 equations, 1 figure, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Dual complex numbers and dual quaternions
  4. Dual quaternion matrix
  5. Dual Complex Adjoint Matrix
  6. The optimal rank-k approximation of dual quaternion Hermitian matrices
  7. The improved two-block coordinate descent method
  8. Experimental Results
  9. Parameter selection and termination condition
  10. Performance of improved two-block coordinate descent method
  11. Conclusions

Key Result

Lemma 1

Let $\hat{A}=A_1+A_2\varepsilon\in \mathbb{DC}^{m\times n}$, $\hat{U}=U_1+U_2\varepsilon\in \mathbb{DC}^{m\times k}$, $\hat{V}=V_1+V_2\varepsilon\in \mathbb{DC}^{m\times k}$, $\hat{\Sigma}= \Sigma_1+\Sigma_2\varepsilon\in \mathbb{DC}^{k\times k}$. If $U_1\Sigma_1V_1^*$ is the best rank-k approximati

Figures (1)

  • Figure 1: The impact of the parameter selection and the termination condition

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 1
  • Lemma 2
  • Definition 2.2
  • Lemma 3
  • Lemma 4
  • Definition 4.1
  • Definition 4.2
  • Lemma 5
  • proof
  • ...and 13 more