  In this paper, we study the low-rank matrix minimization problem, where the
loss function is convex but nonsmooth and the penalty term is defined by the
cardinality function. We first introduce an exact continuous relaxation, that
is, both problems have the same minimzers and the same optimal value. In
particular, we introduce a class of lifted stationary point of the relaxed
problem and show that any local minimizer of the relaxed problem must be a
lifted stationary point. In addition, we derive lower bound property for the
nonzero singular values of the lifted stationary point and hence also of the
local minimizers of the relaxed problem. Then the smoothing proximal gradient
(SPG) algorithm is proposed to find a lifted stationary point of the continuous
relaxation model. Moreover, it is shown that the whole sequence generated by
SPG algorithm converges to a lifted stationary point. At last, numerical
examples show the efficiency of the SPG algorithm.
