Improved two-block coordinate descent method for Pose Graph Optimization Problem under $F^*$-norm
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909824262864896 |
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| author | Chen, Yongjun Zhang, Liping |
| author_facet | Chen, Yongjun Zhang, Liping |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17251 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved two-block coordinate descent method for Pose Graph Optimization Problem under $F^*$-norm Chen, Yongjun Zhang, Liping Numerical Analysis Rings and Algebras 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. |
| title | Improved two-block coordinate descent method for Pose Graph Optimization Problem under $F^*$-norm |
| topic | Numerical Analysis Rings and Algebras |
| url | https://arxiv.org/abs/2407.17251 |