This work, with its two parts, aims at Riemannian optimization on the quaternion Stiefel manifold. Relevant geometric tools have been presented in the first part, while this part demonstrates the applications of the derived tools. To this end, we first propose the model of \(\ell _1\) -norm based quaternion principal component analysis. It is a robustness improvement of the \(\ell _2\) -norm quaternion principal component analysis, which is to alleviate its sensitivity to outliers. The proposed model is formulated as a nonconvex and nonsmooth optimization problem on the quaternion Stiefel manifold. To solve it, we adapt the Riemannian subgradient method built upon the derived geometric tools. Although the objective function is neither geodesically convex nor weakly convex, we still establish global convergence to a critical point for the Riemannian subgradient method with constant or backtracking step-size, and the rate of convergence is shown to be \(O(\frac{1}{\sqrt{k}})\) . Numerical experiments demonstrate the superior performance of the Riemannian subgradient method in comparison with other non-Riemannian algorithms. The robustness of L1-QPCA against outliers is validated by quaternion line-fitting and color image reconstruction experiments.
