Correction: Li, Q.; Liang, S.Y. Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique. Entropy 2017, 19, 421

The authors were not aware of some errors and imprecise descriptions made in the proofreading phase, therefore, we wish to make the following corrections to this paper [...]

Nonparametric Tensor Completion Based on Gradient Descent and Nonconvex Penalty

Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization problem and designs an efficient iterative method to solve it. In each iteration, we not only calculate the missing entries by the aid of data correlation, but consider the low-rank of tensor and the convergence speed of iteration. Our iteration is based on the gradient descent method, and approximates the gradient descent direction with tensor matricization and singular value decomposition. Considering the symmetry of every dimension of a tensor, the optimal unfolding direction in each iteration may be different. So we select the optimal unfolding direction by scaled latent nuclear norm in each iteration. Moreover, we design formula for the iteration step-size based on the nonconvex penalty. During the iterative process, we store the tensor in sparsity and adopt the power method to compute the maximum singular value quickly. The experiments of image inpainting and link prediction show that our method is competitive with six state-of-the-art methods.

Incipient Fault Diagnosis of Rolling Bearings Based on Impulse-Step Impact Dictionary and Re-Weighted Minimizing Nonconvex Penalty Lq Regular Technique

The periodical transient impulses caused by localized faults are sensitive and important characteristic information for rotating machinery fault diagnosis. However, it is very difficult to accurately extract transient impulses at the incipient fault stage because the fault impulse features are rather weak and always corrupted by heavy background noise. In this paper, a new transient impulse extraction methodology is proposed based on impulse-step dictionary and re-weighted minimizing nonconvex penalty Lq regular (R-WMNPLq, q = 0.5) for the incipient fault diagnosis of rolling bearings. Prior to the sparse representation, the original vibration signal is preprocessed by the variational mode decomposition (VMD) technique. Due to the physical mechanism of periodic double impacts, including step-like and impulse-like impacts, an impulse-step impact dictionary atom could be designed to match the natural waveform structure of vibration signals. On the other hand, the traditional sparse reconstruction approaches such as orthogonal matching pursuit (OMP), L1-norm regularization treat all vibration signal values equally and thus ignore the fact that the vibration peak value may have more useful information about periodical transient impulses and should be preserved at a larger weight value. Therefore, penalty and smoothing parameters are introduced on the reconstructed model to guarantee the reasonable distribution consistence of peak vibration values. Lastly, the proposed technique is applied to accelerated lifetime testing of rolling bearings, where it achieves a more noticeable and higher diagnostic accuracy compared with OMP, L1-norm regularization and traditional spectral Kurtogram (SK) method.