Low-Rank Approximation via Generalized Reweighted Iterative Nuclear and Frobenius Norms

Iteratively Capped Reweighting Norm Minimization with Global Convergence Guarantee for Low-Rank Matrix Learning

In recent years, a large number of studies have shown that low rank matrix learning (LRML) has become a popular approach in machine learning and computer vision with many important applications, such as image inpainting, subspace clustering, and recommendation system. The latest LRML methods resort to using some surrogate functions as convex or nonconvex relaxation of the rank function. However, most of these methods ignore the difference between different rank components and can only yield suboptimal solutions. To alleviate this problem, in this paper we propose a novel nonconvex regularizer called capped reweighting norm minimization (CRNM), which not only considers the different contributions of different rank components, but also adaptively truncates sequential singular values. With it, a general LRML model is obtained. Meanwhile, under some mild conditions, the global optimum of CRNM regularized least squares subproblem can be easily obtained in closed-form. Through the analysis of the theoretical properties of CRNM, we develop a high computational efficiency optimization method with convergence guarantee to solve the general LRML model. More importantly, by using the Kurdyka-Łojasiewicz (KŁ) inequality, its local and global convergence properties are established. Finally, we show that the proposed nonconvex regularizer as well as the optimization approach are suitable for different low rank tasks, such as matrix completion and subspace clustering. Extensive experimental results demonstrate that the constructed models and methods provide significant advantages over several state-of-the-art low rank matrix leaning models and methods.

Vibration Source Signal Separation of Rotating Machinery Equipment and Robot Bearings Based on Low Rank Constraint

With the development of industrial robots and other mechanical equipment to a higher degree of automation, mechanical systems have become increasingly complex. This represents a huge challenge for condition monitoring. The separation of vibration source signals plays an important role in condition monitoring and fault diagnosis. The key to the separation method of the vibration source signal is prior knowledge, such as of the statistical features of the vibration source signal, the number of vibration sources, and so forth. However, effective prior knowledge is difficult to obtain in engineering applications. This study found that low rank is a common feature of rotating machinery vibration source signals. To address the problem of the difficulty obtaining the signal feature of a vibration source, the multi-low-rank constrained vibration source signal separation method was proposed. Its advantages and effectiveness have been verified through simulations and experimental tests. Compared with the blind source separation method of independent component analysis (BSS-ICA) and the ensemble empirical mode decomposition (EEMD) methods, it obtained better clustering results and higher signal-to-signal ratio (SSR) values.