  Sparse principal component analysis (PCA), an important variant of PCA,
attempts to find sparse loading vectors when conducting dimension reduction.
This paper considers the nonsmooth Riemannian optimization problem associated
with the ScoTLASS model for sparse PCA which can impose orthogonality and
sparsity simultaneously. A Riemannian proximal method is proposed in the work
of Chen et al. for the efficient solution of this optimization problem. In this
paper, two acceleration schemes are introduced. First and foremost, we extend
the FISTA method from the Euclidean space to the Riemannian manifold to solve
sparse PCA, leading to the accelerated Riemannian proximal gradient method.
Since the Riemannian optimization problem for sparse PCA is essentially
non-convex, a restarting technique is adopted to stabilize the accelerated
method without sacrificing the fast convergence. Second, a diagonal
preconditioner is proposed for the Riemannian proximal subproblem which can
further accelerate the convergence of the Riemannian proximal methods.
Numerical evaluations establish the computational advantages of the proposed
methods over the existing proximal gradient methods on a manifold.
Additionally, a short result concerning the convergence of the Riemannian
subgradients of a sequence is established, which, together with the result in
the work of Chen et al., can show the stationary point convergence of the
Riemannian proximal methods.
