  In the Euclidean setting, the proximal gradient method and its accelerated
variants are a class of efficient algorithms for optimization problems with
decomposable objective. In this paper, we develop a Riemannian proximal
gradient method (RPG) and its accelerated variant (ARPG) for similar problems
but constrained on a manifold. The global convergence of RPG has been
established under mild assumptions, and the O(1/k) is also derived for RPG
based on the notion of retraction convexity. If assuming the objective function
obeys the Rimannian Kurdyka-Lojasiewicz (KL) property, it is further shown that
the sequence generated by RPG converges to a single stationary point. As in the
Euclidean setting, local convergence rate can be established if the objective
function satisfies the Riemannian KL property with an exponent. Moreover, we
have shown that the restriction of a semialgebraic function onto the Stiefel
manifold satisfies the Riemannian KL property, which covers for example the
well-known sparse PCA problem. Numerical experiments on random and synthetic
data are conducted to test the performance of the proposed RPG and ARPG.
