  The paper addresses the problem of optimizing a class of composite functions
on Riemannian manifolds and a new first order optimization algorithm (FOA) with
a fast convergence rate is proposed. Through the theoretical analysis for FOA,
it has been proved that the algorithm has quadratic convergence. The
experiments in the matrix completion task show that FOA has better performance
than other first order optimization methods on Riemannian manifolds. A fast
subspace pursuit method based on FOA is proposed to solve the low-rank
representation model based on augmented Lagrange method on the low rank matrix
variety. Experimental results on synthetic and real data sets are presented to
demonstrate that both FOA and SP-RPRG(ALM) can achieve superior performance in
terms of faster convergence and higher accuracy.
