  In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized
matrix minimization problems. In particular, we first introduce a class of
first-order stationary points for them, and show that the first-order
stationary points introduced in [11] for an SPQN regularized $vector$
minimization problem are equivalent to those of an SPQN regularized $matrix$
minimization reformulation. We also show that any local minimizer of the SPQN
regularized matrix minimization problems must be a first-order stationary
point. Moreover, we derive lower bounds for nonzero singular values of the
first-order stationary points and hence also of the local minimizers of the
SPQN regularized matrix minimization problems. The iterative reweighted
singular value minimization (IRSVM) methods are then proposed to solve these
problems, whose subproblems are shown to have a closed-form solution. In
contrast to the analogous methods for the SPQN regularized $vector$
minimization problems, the convergence analysis of these methods is
significantly more challenging. We develop a novel approach to establishing the
convergence of these methods, which makes use of the expression of a specific
solution of their subproblems and avoids the intricate issue of finding the
explicit expression for the Clarke subdifferential of the objective of their
subproblems. In particular, we show that any accumulation point of the sequence
generated by the IRSVM methods is a first-order stationary point of the
problems. Our computational results demonstrate that the IRSVM methods
generally outperform some recently developed state-of-the-art methods in terms
of solution quality and/or speed.
