The Research Of Robust Principal Component Analysis Method

In many domains,such as pattern recognition and machine learning,the dimensionality of data could be more than thousands.Principal Component Analysis?PCA?is a widely used tool for dimensionality reduction and feature extraction in the field of computer vision.Three problems can be solved by reducing the dimensions of high-dimensional data.First,it will cause the curse of dimensionality and performance degradation by directly handling high-dimensionality datain the case of high-dimensional data.PCA is one of the most classical method to alleviate the dimensional disaster.Second,PCA can minimize the loss of information,while compressing the data.Third,the dimensionality-reduced data can be more easily analyzed and understood through dimensionality reduction using the PCA method.However,Traditional PCA is sensitive to outliers which are common in empirical applications.Therefore,in recent years,massive efforts have been made to improve the robustness of PCA.However,many emerging PCA variants developed in the direction have some weaknesses.First,to estimate data mean from sample set with outliers by averaging is usually biased.Second,if some elements of a sample are disturbed,to extract principal components?PCs?by directly projecting data with transformation matrix causes incorrect mapping of sample to its genuine location in low-dimensional feature subspace.Third,few of them pay attention to the matrix norm.Fourth,most robust methods cannot maintain the good characteristics of 2-DPCA,which are important for learning algorithms,such as rotation invariance.The above problems bring great challenges to the performance improvement of robust PCA.To improve the robustness of PCA,this thesis revisited the model of the robust PCA and proposes two new robust PCA methods.The main research work is as follows:1.This thesis revisited the robust PCA method and finds that most of the existing robust PCA methods are biased in calculating the sample mean and the low-dimensional representation of the sample.Therefore,this thesis re-explains the objective function of the robust PCA method to improve the robustness of the algorithm.In the new model,the sample mean is used as an optimization variable.At the same time,this thesis proposes a method for estimating the true position of an image in the low-dimensional feature subspace.However,the low-dimensional representation of the sample of the most existing PCA is calculated by direct projection.This thesis discusses the applicable range of the newly proposed objective function and conducts theoretical and experimental analysis of its effectiveness.Meanwhile,a novel framework is developed to handle unseen sample.Moreover,it is known that 2D projection based PCA methods can exist a weakness in actual application.Therefore,a new robust method with two stages is proposed to alleviate this problem.2.To solve the structural noise,a novel PCA based on nuclear norm is proposed under the new model which can help improve the robustness by coping the contaminated features in image samples.It is generally known that when there are outliers,the distance metric will seriously affect the effectiveness of the algorithm.Moreover,the structured noise?e.g.block occlusion?causes that the error image is of low rank.Usually,the low-rank function is difficult to solve since it is a non-convex function.Nuclear norm minimization is used instead of minimizing rank due to nuclear norm is the convex envelope of matrix rank.N-PCA uses nuclear norm to measure the reconstruction error,which can make full use of the spatial structure information and calculate a robust projection matrix for outliers.In addition,N-PCA also uses new models to evaluate the sample mean and the low-dimensional representation of the data to improve the robustness of the algorithm.3.This thesis extends F-2-DPCA to a generalized robust distance metric learning method,namely L_2,p-2-DPCA.In L_2,p-2-DPCA,the the reconstruction error in spatial dimensions is measured by F-norm,while the summation of all samples uses pL-norm.To solve L_2,p-2-DPCA,an iterative algorithm is presented,which has a closed-form solution in each iteration.Compared to most existing robust PCA methods,L_2,p-2-DPCA has the following advantages.First,our method is robust to outliers due to the facts that L_2,p-norm weakens the inuence of large distance.Second,L_2,p-2-DPCA proposes an effective way to automatically estimate sample mean.Third,L_2,p-2-DPCA preserves desirable property of 2-DPCA that is rotational invariance.Moreover,this thesis also demonstrates that 2-DPCA and F-2-DPCA are two special cases of our model.