Nonconvex Model For Robust Principal Component Analysis With Its Applications

How to effectively recover the low-rank component from the partial observations which contain outliers or noisy corruptions,the research on this problem has been widely used in modern life,including various fields such as machine learning,data mining and image processing,etc.For example,the shopping sites need to take their users' interest into consideration when they correctly recommend goods from massive commodities.Driven by actual demand,the theory of the model for principle component analysis(PCA)is developing gradually,which is one of the mainstream methods of data analysis.However,the model for PCA lacks robustness and sensitivity to outliers or non-Gaussian noises,which limits its application in real scenarios.In order to deal with this problem,the model for robust PCA(RPCA)has become a hot issue,which is considered as the extension of PCA.Compared with PCA,the model for RPCA not only effectively recover the low-rank component of observed data,but also get the sparse component.Candes[14]et al.proposed the convex optimization model of RPCA was obtained by using the convex envelopes of the l0-norm and rank function(l1-norm and nuclear norm).In this paper,a new nonconvex model for RPCA is proposed to achieve better performance,and a detailed algorithm is given.The main work of the paper is given as follows:Firstly,this paper introduces the lp-norm and proposes the nonconvex model for RPCA based on the lp-norm(lp-RPCA).Aiming at the defects of the convex relaxation of RPCA,the new nonconvex model used the lp-norm of a matrix to approximate the l0-norm,and the rank function is approximated by the lp-norm of the singular value vector.Compared with the convex optimization model for RPCA based on the l1-norm and nuclear norm,the nonconvex model for lp-RPCA can approximate the real model.Secondly,the alternating direction method of multipliers is used to solve the nonconvex model for lp-RPCA.In the process of solving this problem,the result of the lp-norm penalized least squares problem for matrix completion proposed by Marjanovic[1]et al.is used,and the algorithm is given in detail in this paper.Finally,this paper applies the new nonconvex model for lp-RPCA to synthetic data and real-world datasets(including gray image denoising,shadow and specularity removal form face images,video background modeling).These experimental results show that the nonconvex model for lp-RPCA can recover the low-rank and sparse components of the observation matrix successfully.Compared with other methods,numerical experiments show that the proposed nonconvex model for RPCA is superior to other models.