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.

本文通过建立Schatten-p norm与factor matrices中的Schatten-p1和Schatten-p2 norm之间的等价性，进一步扩展到多个因子矩阵中，表明其所有的因子norm都可以是任意p的凸光滑的，从而利用凸性提出加速的proximal alternating linearized minimization algorithm，并在合成和实际数据集上通过矩阵补全任务展示了其优越性能。

一种蕴含 Schatten-$p$ 范数的凸代理函数