  Pose graph optimization (PGO) is a well-known technique for solving the
pose-based simultaneous localization and mapping (SLAM) problem. In this paper,
we represent the rotation and translation by a unit quaternion and a
three-dimensional vector, and propose a new PGO model based on the von
Mises-Fisher distribution. The constraints derived from the unit quaternions
are spherical manifolds, and the projection onto the constraints can be
calculated by normalization. Then a proximal linearized Riemannian alternating
direction method of multipliers (PieADMM) is developed to solve the proposed
model, which not only has low memory requirements, but also can update the
poses in parallel. Furthermore, we establish the iteration complexity of
$O(1/\epsilon^{2})$ of PieADMM for finding an $\epsilon$-stationary solution of
our model. The efficiency of our proposed algorithm is demonstrated by
numerical experiments on two synthetic and four 3D SLAM benchmark datasets.
