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Low-rank modeling has many important applications in computer vision and machine learning. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better empirical performance. However, the resulting optimization problem is much more challenging. Recent state-of-the-art requires an expensive full SVD in each iteration. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, the singular values obtained from the proximal operator can be automatically threshold. This allows the proximal operator to be efficiently approximated by the power method. We then develop a fast proximal algorithm and its accelerated variant with inexact proximal step. It can be guaranteed that the squared distance between consecutive iterates converges at a rate of $O(1/T)$O(1/T), where $T$T is the number of iterations. Furthermore, we show the proposed algorithm can be parallelized, and the resultant algorithm achieves nearly linear speedup w.r.t. the number of threads. Extensive experiments are performed on matrix completion and robust principal component analysis. Significant speedup over the state-of-the-art is observed. [ABSTRACT FROM AUTHOR]

Matrix and tensor completion aim to recover a low-rank matrix / tensor from limited observations and have been commonly used in applications such as recommender systems and multi-relational data mining. A state-of-the-art matrix completion algorithm is Soft-Impute, which exploits the special "sparse plus low-rank" structure of the matrix iterates to allow efficient SVD in each iteration. Though Soft-Impute is a proximal algorithm, it is generally believed that acceleration destroys the special structure and is thus not useful. In this paper, we show that Soft-Impute can indeed be accelerated without comprising this structure. To further reduce the iteration time complexity, we propose an approximate singular value thresholding scheme based on the power method. Theoretical analysis shows that the proposed algorithm still enjoys the fast $O(1/T^2)$O(1/T2) convergence rate of accelerated proximal algorithms. We also extend the proposed algorithm to tensor completion with the scaled latent nuclear norm regularizer. We show that a similar "sparse plus low-rank" structure also exists, leading to low iteration complexity and fast $O(1/T^2)$O(1/T2) convergence rate. Besides, the proposed algorithm can be further extended to nonconvex low-rank regularizers, which have better empirical performance than the convex nuclear norm regularizer. Extensive experiments demonstrate that the proposed algorithm is much faster than Soft-Impute and other state-of-the-art matrix and tensor completion algorithms. [ABSTRACT FROM AUTHOR]