Study On Low-rank Reprsentation Alogorithms And Applications Based On Non-convex Approximation

This paper mainly studies the Low Rank Representation model and its algorithm application.In recent years,the theory and application of Low Rank Representation has attracted the attention of many scholars,and has achieved remarkable results in image processing and data analysis.How to recover the low rank structure hidden in high-dimensional data is the key to solve the problems in various fields by using low rank representation.The restoration of the matrix in the low rank representation model is mainly dependent on the low rank constraint on the matrix.Because the rank function is a discrete function,the model of minimizing the rank is a NP-hard problem.Although the existing model uses the kernel norm,the minimum convex envelope of the rank function as the convex rank approximation function,it has obtained a certain degree.But in many practical applications,nuclear norm has shown the existence of approximate matrix rank defect.In this paper,based on the more accurate non convex rank approximation function of the low rank matrix,two low rank representation optimization models based on the non-convex rank approximation are proposed for the existing low rank representation model,and the correlation algorithm is used to solve it.Finally,it is applied to the subspace clustering.The main work is summarized as follows:First,the relevant theoretical knowledge about the low rank representation model is introduced,including subspace clustering,low rank representation model,alternating direction multiplier method,non-convex approximation function and so on.First,the application background of low rank representation model,namely subspace clustering problem,is introduced.Secondly,the general model and research development of low rank representation are summarized,and the alternating direction multiplier method is used to solve it,and the algorithm framework of the alternating direction multiplier method is given.Finally,this paper analyzes the shortage of the kernel norm as the rank approximation of the matrix in the low rank representation model,and gives the design conditions that the non-convex rank approximation function should be satisfied.Second,in view of the problem of the approximate deviation of the matrix rank of the standard kernel norm in the low rank representation model,the low rank matrix is decomposed to the low rank matrix in the existing low rank representationmodel,and the low rank matrix is decomposed into three small scale matrices,and the logarithmic determinant function is used as the non-convex rank approximation of the low rank matrix.Function,a low rank representation model based on matrix decomposition and logarithmic determinant function is established.By using the Linearization technique,the non-convex optimization model is solved by alternating direction multiplier method and each variable is updated alternately,and the optimal solution of each sub problem in the model is obtained.The affine matrix is constructed by constructing the low rank representation matrix,and applied to the subspace clustering.The matrix decomposition is used to simplify the singular value decomposition of the low rank matrix,and then reduce the computational complexity and improve the efficiency of the algorithm.The logarithmic determinant function is superior to the traditional kernel norm as the non-convex approximate term of the rank of the matrix.The accuracy of the algorithm is improved,and a more accurate low rank matrix recovery effect is obtained.Finally,simulation data and real data experiments show that the proposed method is more limited and robust than the existing methods.Third,aiming at face recognition problem in image recognition,we use low rank representation model and subspace clustering.In the low rank representation model,a new non convex rank approximation function based on the exponential function is used,and a low rank representation model based on the non-convex rank approximate exponential function is established.The exponential function is more accurate than the traditional kernel function.Compared with the traditional kernel function,the approximate effect of the exponential function is obviously better than that of the kernel when the single singular value is large.Using exponential function to do the non-convex rank approximation of coefficient lower rank matrix,we can find the low rank representation matrix of potential structure in the data more clearly.The augmented Lagrange multiplier method is used to solve the transformed model.The affine matrix is constructed for the obtained low rank representation coefficient matrix.Finally,the affine matrix obtained by the low rank representation model is used for subspace clustering.Because more accurate rank approximation is used,which helps to recover more accurately the low rank matrix,and further helps to cluster later.The data experiment shows that this method has lower clustering error rate for multi object clustering problem,which shows the robustness of this method,and has obvious advantages compared with other existing subspace clustering methods based on spectral clustering,which shows the effectiveness of this method.Finally,the main content of the article is summed up,further analysis of the shortcomings of the algorithm proposed in this paper,and the direction of the next step of the research.