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

Fast Alternating Projections on Manifolds Based on Tangent Spaces