Large-Scale Affine Matrix Rank Minimization With a Novel Nonconvex Regularizer


The goal of low-rank minimization is to recover a matrix with the lowest possible rank while still satisfying the constraints of a linear system. It is utilized in a variety of data analysis and Machine Learning applications, including Signal Processing, video denoising, and recommender systems, to name a few. The minimization of the nuclear norm is the most common approach taken to deal with it. However, this method does not take into account the disparity that exists between singular values of the target matrix. In order to find a solution to this problem, nonconvex low-rank regularizers have seen widespread application. Unfortunately, the methods that are currently used suffer from a variety of drawbacks, including inefficiency and inaccurate results. This article makes a suggestion for a flexible model that uses an original nonconvex regularizer as a means of resolving issues of this nature. A model with these characteristics not only encourages low rankness, but it also allows for much faster and more accurate problem solving. With it, the initial low-rank problem can be equivalently transformed into the resulting optimization problem under the rank restricted isometry property (rank-RIP) condition. This is possible because of the existence of the rank restricted isometry property. After that, Nesterov's rule and inexact proximal strategies are adopted to achieve a novel algorithm that is highly efficient in solving this problem at a convergence rate of O(1/K), where K is the iterate count. This algorithm achieves a convergence rate of O(1/K). In addition to that, the asymptotic convergence rate is also subjected to a stringent analysis by making use of the Kurdyka-ojasiewicz (KL) inequality. In addition, we apply the proposed optimization model to common low-rank problems, such as the completion of tensors and matrices, as well as robust principal component analysis (RPCA). The proposed model outperforms state-of-the-art models in terms of accuracy and efficiency, according to exhaustive empirical studies regarding data analysis tasks such as synthetic data analysis, image recovery, personalized recommendation, and background subtraction.

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