  Recovering a large matrix from limited measurements is a challenging task
arising in many real applications, such as image inpainting, compressive
sensing and medical imaging, and this kind of problems are mostly formulated as
low-rank matrix approximation problems. Due to the rank operator being
non-convex and discontinuous, most of the recent theoretical studies use the
nuclear norm as a convex relaxation and the low-rank matrix recovery problem is
solved through minimization of the nuclear norm regularized problem. However, a
major limitation of nuclear norm minimization is that all the singular values
are simultaneously minimized and the rank may not be well approximated
\cite{hu2012fast}. Correspondingly, in this paper, we propose a new multi-stage
algorithm, which makes use of the concept of Truncated Nuclear Norm
Regularization (TNNR) proposed in \citep{hu2012fast} and Iterative Support
Detection (ISD) proposed in \citep{wang2010sparse} to overcome the above
limitation. Besides matrix completion problems considered in
\citep{hu2012fast}, the proposed method can be also extended to the general
low-rank matrix recovery problems. Extensive experiments well validate the
superiority of our new algorithms over other state-of-the-art methods.
