Improved Robust Principal Component Analysis Model And Its Applications

In the wake of developments in science and technology, data collection is getting easier. And we have come to the age of big data. Large data sets contain large amounts of information, but also bring noises inevitably. So how to find out valuable information from the corrupted data becomes a problem that urgently to be resolved. Principal component analysis is one of the popular and mature methods in computational data analysis. However, this method is sensitive to outliers and lack of robustness so that its applicability is limited in real scenarios. John Wright et al. proposed the RPCA model, which overcame the PCA'S shortcoming by minimizing the nuclear norm and L1-norm. Based on the robust principle component analysis model, we propose a new model to generate the row sparsity and give the relevant algorithm. But beyond that, we also give two algorithms to solve the model quickly. We begin studying from the following three important aspects:1. We introduce the L21-norm to the RPCA model to describe the data structure?In the RPCA model, the error matrix should be sparse, and it uses the L1-norm to achieve the sparsity, without considering the structure of the data itself. The L21-norm based on the loss function is robust to outliers in data points and L21-norm regularization is more reasonable when the data is grossly corrupted. Especially in the problem of feature selection,the L21-norm selects features across all data points with joint sparsity.2. Two new models are proposed to solve the large problems quickly. In order to obtain the low-rank structure,RPCA model minimizes the nuclear norm of the matrix. But the most expensive computational task required by nuclear-norm minimization algorithms is to perform the singular value decomposition at each iteration, which becomes increasingly costly as the matrix dimensions grow. To further improve the scalability of solving the large-scale problems, we adopt two ways to avoid nuclear norm minimization: one is the non-convex matrix factorization approaches, using a matrix product to achieve the low-rank constraint. Another is using variational definition of the nuclear norm.3. Applying the new models to the background removing and face illumination. These improved models have expanded the scope of the original model. So we apply these new models to the real scenarios, compare them with each other and find that the new models not only can get more satisfactory solutions but also take less time than the original.