Research On Low Rank Representation Based On Manifold Optimization

With the increasing development of the Internet and information technology,a large amount of image data is continuously generated.How to label these data quickly and efficiently is a big problem.Cluster analysis is an unsupervised learning method that can divide data samples into different categories according to the similarity principle without using any prior information.Among many unsupervised learning methods,the subspace clustering method is favored by researchers because of its powerful processing effect on high-dimensional data and its high interpretability.Low-rank representation is one of the most representative models in subspace clustering algorithms,and its clustering effect is remarkable,especially when dealing with challenging clustering problems.However,it still suffers from poor performance in the face of nonlinear structured data,and it is difficult to preserve the local geometric structure of the data.In response to these problems,this paper analyzes the shortcomings of existing models based on low-rank representation models,and designs three new low-rank representation models to improve clustering performance.The main research of this paper is as follows:Aiming at the problem that the traditional low-rank representation method is difficult to preserve the nonlinear structure of the data,the real data often has a nonlinear structure.Therefore,a nonlinear low-rank representation method by estimating optimal transformations(EOTLRR)is proposed,which map data from the original space to a new linear space to increase the correlation between samples that may belong to the same class,which is conducive to better preserve the global structure of the data.The objective function established by estimating the optimal transformation can maximize the p-ky norm of the data sample covariance matrix.In addition,the alternative conditional expectation method is used to complete the estimation of the optimal transformation.In most LRR-based methods,the two steps of data reduction and learning low-rank representation coefficients are carried out separately,which cannot guarantee the adaptability of representation coefficients to the original data space.Therefore,an adaptive dimensionality reduction low-rank representation with adaptive dimensionality reduction via manifold optimization(LRRARD)is proposed,which integrates the data dimensionality reduction technology and the learning of low-rank representation coefficients into an overall framework.In this model,a low-dimensional projection matrix is introduced to find the optimal projection fitting the original data space,and the projection matrix and the representation coefficient are optimized together and the optimal value is obtained at the same time.In addition,the generalized Stiefel manifold optimization method is utilized to optimize the projection matrix.Although LRRARD successfully integrates data dimensionality reduction technology into LRR,it does not consider the local structure problem that LRR is difficult to retain data.Therefore,in order to further improve the clustering performance,a new low-rank representation with projection distance regularization via manifold optimization for clustering(PDRLRR)is proposed on the basis of LRRARD.The projection distance regularization term is introduced,and the Schatten-p norm is used to replace the nuclear norm to solve the rank minimization problem,thus finding a more accurate low-rank solution.The projection distance regularization term can simultaneously keep the global and local geometric structure of the data,making the expression coefficient more complete to retain the structural information between the data samples,so the similarity matrix constructed is also more reasonable.By using the Schatten-p norm to solve the rank minimization problem to obtain a more accurate low-rank solution,it helps to improve the discriminant information and robustness of the representation matrix.