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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 associatedwith the ScoTLASS model for sparse PCA which can impose orthogonality andsparsity simultaneously. A Riemannian proximal method is proposed in the workof Chen et al. for the efficient solution of this optimization problem. In thispaper, two acceleration schemes are introduced. First and foremost, we extendthe FISTA method from the Euclidean space to the Riemannian manifold to solvesparse PCA, leading to the accelerated Riemannian proximal gradient method.Since the Riemannian optimization problem for sparse PCA is essentiallynon-convex, a restarting technique is adopted to stabilize the acceleratedmethod without sacrificing the fast convergence. Second, a diagonalpreconditioner is proposed for the Riemannian proximal subproblem which canfurther accelerate the convergence of the Riemannian proximal methods.Numerical evaluations establish the computational advantages of the proposedmethods over the existing proximal gradient methods on a manifold.Additionally, a short result concerning the convergence of the Riemanniansubgradients of a sequence is established, which, together with the result inthe work of Chen et al., can show the stationary point convergence of theRiemannian proximal methods.

An Extension of Fast Iterative Shrinkage-thresholding to Riemannian Optimization for Sparse Principal Component Analysis