  In this paper, we develop a new alternating projection method to compute
nonnegative low rank matrix approximation for nonnegative matrices. In the
nonnegative low rank matrix approximation method, the projection onto the
manifold of fixed rank matrices can be expensive as the singular value
decomposition is required. 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 the alternating
projections onto the tangent spaces of the fixed rank matrices manifold and the
nonnegative matrix manifold, converge linearly to a point in the intersection
of the two manifolds where the convergent point is sufficiently close to
optimal solutions. This convergence result based inexact projection onto the
manifold is new and is not studied in the literature. Numerical examples in
data clustering, pattern recognition and hyperspectral data analysis are given
to demonstrate that the performance of the proposed method is better than that
of nonnegative matrix factorization methods in terms of computational time and
accuracy.
