  Affine matrix rank minimization problem is a fundamental problem with a lot
of important applications in many fields. It is well known that this problem is
combinatorial and NP-hard in general. In this paper, a continuous promoting low
rank non-convex fraction function is studied to replace the rank function in
this NP-hard problem. Inspired by our former work in compressed sensing, an
iterative singular value thresholding algorithm is proposed to solve the
regularization transformed affine matrix rank minimization problem. For
different $a>0$, we could get a much better result by adjusting the different
value of $a$, which is one of the advantages for the iterative singular value
thresholding algorithm compared with some state-of-art methods. Some
convergence results are established and numerical experiments show that this
thresholding algorithm is feasible for solving the regularization transformed
affine matrix rank minimization problem. Moreover, we proved that the value of
the regularization parameter $\lambda>0$ can not be chosen too large. Indeed,
there exists $\bar{\lambda}>0$ such that the optimal solution of the
regularization transformed affine matrix rank minimization problem is equal to
zero for any $\lambda>\bar{\lambda}$. Numerical experiments on matrix
completion problems show that our method performs powerful in finding a
low-rank matrix and the numerical experiments about image inpainting problems
show that our algorithm has better performances than some state-of-art methods.
