Study Of Robust Principal Component Analysis And Its Applications

Principal Component Analysis?PCA?has been widely applied in data representation,data denoising,and discrimination.During the past decades,research on PCA has achieved great progress.With the rapid development of data collection technology,data collected in real-world applications are usually of extremely high dimension,and are unavoidable to contain noises or outliers,which makes the distribution of the collected data deviate from the real distribution.PCA measures similarity between data points with squared Euler distance and hence emphasizes the influence of data with long distance.Therefore,when employ-ing PCA to analyze these data,it usually suffers from performance degradation and poor robustness.To tackle this problem,numerous robust PCA methods are developed.Some methods replace squared Euler distance with other distance measurement metrics for better robustness.These methods usually neglect the linear relationship between variance and re-construction error,which makes low-dimensional data representation inaccurate.Besides,most of them fail to attain robustness and rotational invariance simultaneously.Differen-t from these methods,low-rank PCA aims at denoising and adopts transductive learning.However,it cannot solve out-of-sample problem,which greatly limits its real-world applica-tion.Aiming at these problems,we conduct in-depth research on two kinds of robust PCA methods,summarized as follows:1.To solve the problems of robust PCA based on robust distance measurement that it cannot attain robustness and rotational invariance simultaneously,we propose?_2,-norm based PCA?L2p-PCA?.L2p-PCA is the extension of?₂-norm based PCA and achieves ro-bustness meanwhile it guarantees rotational invariance.Existing robust PCA methods based on robust distance measurement fail to minimize reconstruction error and maximize vari-ance simultaneously.Besides,the obtained low-dimensional projection lacks the constraint on reconstruction error or variance,which results in inaccuracy of low-dimensional repre-sentation.To overcome this weakness,we embed reconstruction error and low-dimensional representation into the objective function and then develop a novel robust PCA model,i.e.angle PCA.It effectively improves robustness and achieves rotational invariance.Besides,its solution can fulfill the two constraints of reconstruction error minimization and variance maximization.Experimental results demonstrate that the proposed algorithm significantly improves the robustness of one-dimensional PCA.2.The aforementioned methods need to transfer image matrix into vector when pro-cessing images,which leads to geometric information missing.Existing tow-dimensional PCA?2DPCA?methods based on robust distance measurement metrics usually adopt op-timal mean based on squared-norm to centralize samples rather than the optimal mean based on adopted robust distance measurement metric.This results in inaccuracy of cen-tralized results and reduces robustness.To address this problem,we propose optimal mean2DPCA with-norm minimization,i.e.,OMF-2DPCA.This method employs-norm to measure data similarity and calculate optimal mean with-norm to improve robustness.Besides,we extend the proposed one-dimensional robust PCA method and proposed opti-mal mean 2DPCA with?_2,1-norm minimization?OM L21-2DPCA?and optimal mean an-gle 2DPCA?OM Angle 2DPCA?.Compared with existing 2DPCA methods,the proposed methods adopt robust distance measurement metrics and use optimal mean to centralize samples,which helps achieve better resistance to outliers/noises.Experiments on several face databases added with noises are conducted,and the experimental results demonstrate that the proposed methods greatly improve the robustness of two-dimensional robust PCA and achieve fast convergence.3.Robust Principal Component Analysis?RPCA?is incapable of handling new sam-ples out of training dataset.To address this issue,we propose double robust PCA method?DRPCA?,which simultaneously adds low-rank constraints to both clean data samples ob-tained with low-rank sparse matrix decomposition and the linear projection between clean dataset and noisy dataset.In this way,DRPCA improves denoising results on training sam-ples and the learned projection can be utilized for denoising on new data samples out of training dataset.Compared with existing low-rank RPCA,DRPCA employs reconstruction error to obtain intrinsic geometric structure of data and further embeds it into the RPCA model adaptively.The obtained low-rank representation well keeps the intrinsic geometric structure and can better characterize distribution of data of different categories.Finally,with a series of experiments of denoising,foreground extraction,and clustering tasks,the paper successfully demonstrates the superiority of DRPCA.4.We conduct in-depth research on robust PCA based on robust distance measurement and RPCA as summarized above.Robust PCA based on robust distance measurement only considers the optimal projection for data dimension reduction,but it cannot well remove data noises.Although RPCA can remove data noises/outliers,it cannot reduce data dimension or conduct robust feature extraction.Therefore,it fails to handle out-of-sample problem.Aiming at these problems,we combine these two methods and propose enhanced robust PCA?ERPCA?.Compared with existing methods,it can effectively eliminate noises/outliers from images before processing for better robustness.Besides,it can also conduct robust dimension reduction on data from training set and testing data out of training set.Finally,experimental results on real world datasets show that the correctness and validity of the proposed algorithm.The papers starts from low-dimensional representation and low-rank representation of PCA,studies robust classification and data denosing,and finally solves a series of challeng-ing problems.