In Riemannian optimization problems, commonly encountered manifolds are $d$-dimensional matrix manifolds whose tangent spaces can be represented by $d$-dimensional linear subspaces of a $w$-dimensional Euclidean space, where $w > d$. Therefore, representing tangent vectors by $w$-dimensional vectors has been commonly used in practice. However, using $w$-dimensional vectors may be the most natural but may not be the most efficient approach. A recent paper, [Mathematical Programming Series A, 150:2, pp. 179-216, 2014], proposed using $d$-dimensional vectors to represent tangent vectors and showed its benefits without giving detailed implementations for commonly encountered manifolds. In this paper, we discuss the implementations of using $d$-dimensional vectors to represent tangent vectors for the Stiefel manifold, the Grassmann manifold, the fixed-rank manifold and the manifold of positive semidefinite matrices with rank fixed. A Riemannian quasi-Newton method for minimizing the Brockett cost function is used to demonstrate the performance of the $d$-dimensional representation.
Universite catholique de Louvain; Tech. report-UCL-INMA-2016.08; May 2016