  The Schatten-$p$ norm ($0<p<1$) has been widely used to replace the nuclear
norm for better approximating the rank function. However, existing methods are
either 1) not scalable for large scale problems due to relying on singular
value decomposition (SVD) in every iteration, or 2) specific to some $p$
values, e.g., $1/2$, and $2/3$. In this paper, we show that for any $p$, $p_1$,
and $p_2 >0$ satisfying $1/p=1/p_1+1/p_2$, there is an equivalence between the
Schatten-$p$ norm of one matrix and the Schatten-$p_1$ and the Schatten-$p_2$
norms of its two factor matrices. We further extend the equivalence to multiple
factor matrices and show that all the factor norms can be convex and smooth for
any $p>0$. In contrast, the original Schatten-$p$ norm for $0<p<1$ is
non-convex and non-smooth. As an example we conduct experiments on matrix
completion. To utilize the convexity of the factor matrix norms, we adopt the
accelerated proximal alternating linearized minimization algorithm and
establish its sequence convergence. Experiments on both synthetic and real
datasets exhibit its superior performance over the state-of-the-art methods.
Its speed is also highly competitive.
