Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better recovery performance. However, the resultant optimization problem is much more challenging. A very recent state-of-the-art is based on the proximal gradient algorithm. However, it requires an expensive full SVD in each proximal step. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows the use of power method to approximate the SVD efficiently. Besides, the proximal operator can be reduced to that of a much smaller matrix projected onto this leading subspace. Convergence, with a rate of O(1/T) where T is the number of iterations, can be guaranteed. Extensive experiments are performed on matrix completion and robust principal component analysis. The proposed method achieves significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the traditional nuclear norm regularizer.

本文提出了一种基于截断的异性低秩正则化方法，通过使用功率方法逼近奇异值分解以提高计算效率，相比于传统核范数正则化方法，实验结果表明所提出的方法在矩阵补全领域有更快的速度和更高的准确率。

快速低秩矩阵学习与非凸正则化