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

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.