Efficient Two-Block Coordinate Descent Methods for Pose Graph Optimization Problem

Simultaneous localization and mapping (SLAM) problem is very important in robotics research. In this paper, based on $F^*$-norm of dual quaternion matrices, we establish the pose graph optimization (PGO) model for SLAM and propose an improved two-block coordinate descent method to solve the PGO model. We show that one block has a closed-form solution and another is the optimal rank-one approximation of dual quaternion Hermitian matrices under $F^*$-norm. We derive an explicit solution for the rank-one approximation and present an efficient algorithm to compute the optimal rank-one approximation. 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, the performances of PGO with $F^*$-norm outperforms the traditional one with F-norm, with respect to its ability to more faithfully capture the magnitude of the dual parts of dual quaternion matrices.