  In this paper, we study alternating projections on nontangential manifolds
based on the tangent spaces. The main motivation is that the projection of a
point onto a manifold can be computational expensive. We propose to use the
tangent space of the point in the manifold to approximate the projection onto
the manifold in order to reduce the computational cost. We show that the
sequence generated by alternating projections on two nontangential manifolds
based on tangent spaces, converges linearly to a point in the intersection of
the two manifolds where the convergent point is close to the optimal solution.
Numerical examples for nonnegative low rank matrix approximation and low rank
image quaternion matrix (color image) approximation, are given to demonstrate
that the performance of the proposed method is better than that of the
classical alternating projection method in terms of computational time.
