  Low rank matrix approximation is a popular topic in machine learning. In this
paper, we propose a new algorithm for this topic by minimizing the
least-squares estimation over the Riemannian manifold of fixed-rank matrices.
The algorithm is an adaptation of classical gradient descent within the
framework of optimization on manifolds. In particular, we reformulate an
unconstrained optimization problem on a low-rank manifold into a differential
dynamic system. We develop a splitting numerical integration method by applying
a splitting integration scheme to the dynamic system. We conduct the
convergence analysis of our splitting numerical integration algorithm. It can
be guaranteed that the error between the recovered matrix and true result is
monotonically decreasing in the Frobenius norm. Moreover, our splitting
numerical integration can be adapted into matrix completion scenarios.
Experimental results show that our approach has good scalability for
large-scale problems with satisfactory accuracy
