| With the rapid development of the Internet,cloud computing and other advanced technologies,the data people obtain are increasingly showing high-dimensional characteristics.How to obtain the desired effective information from these highdimensional data is a topic of great significance.Subspace clustering is a clustering analysis method that can effectively deal with high-dimensional data,and it has been widely used in the fields of computer vision,image processing and pattern recognition.Low Rank Representation(LRR)is a classical self-representation-based subspace clustering method,but it suffers from serious problems such as insufficient data sampling and severe corruption.Latent Low Rank Representation(LLRR)overcomes these difficulties by considering both observed and unobserved data.However,LLRR uses convex nuclear norm to achieve the low-rank constraint,ignoring the differences among different singular values,which leads to the fact that the learned low-rank representation coefficient matrix cannot accurately describe the class structure of the data.Numerous studies and experiments have demonstrated that non-convex functions can more accurately approximate the rank function.However,there are usually no closed-form solutions and large time costs in solving these non-convex regularizations-based models.To overcome the above problems,based on non-convex Schatten-p norm and LLRR,this thesis proposes two new models for subspace clustering,and designs effective iterative optimization algorithms to solve the models.The main work is as follows:(1)Aiming at the problem that LLRR uses convex nuclear norm to treat all singular values equally,ignoring the differences between different singular values,so that the learned representation coefficient matrix cannot accurately describe the class structure of the data,this thesis uses non-convex weighted Schatten-p norm(WSN)to approximate the rank function and designs a robust subspace clustering based on latent low-rank representation with weighted Schatten-p norm(WSN-LLRR)The proposed non-convex optimization problem is solved by an efficient iterative optimization solution algorithm using the alternating direction method of multipliers(ADMM)framework.WSN-LLRR is extended from LLRR and considers the unobserved data,which can effectively overcome the problem of insufficient data sampling and improve the robustness of the algorithm against noise and corruption.It also explicitly considers the difference between different singular values and can more accurately induce the low-rank structure of the representation coefficient matrix.The similarity matrix for spectral clustering is constructed by exploring the angular information of the principal directions of the lowrank representation to further improve the subspace clustering performance.Extensive experimental results verify the effectiveness of the proposed method.(2)The WSN thresholding function can only obtain approximate solutions and may cause more time consumption.This thesis uses the non-convex transformed Schatten-1(TS1)function to approximate the rank of the representation coefficient matrix,and proposes robust subspace clustering based on latent low-rank representation with TS1function(TS1-LLRR).An effective iterative optimization algorithm is proposed by using ADMM framework.On the one hand,TS1-LLRR is extended from LLRR,which has good robustness.On the other hand,TS1 function can estimate the rank function well and its threshold function has closed-form analytical solutions for all parameter values α∈(0,+∞),therefore,each subproblem obtained in the optimization process of TS1-LLRR model has closed-form solutions,which ensures the accuracy of model’s solutions and the solving time is relatively short.The ability to reveal the intrinsic correlation between data points is further enhanced by constructing similarity matrix through some post-processing techniques on the low-rank representation.Numerous experimental results on image clustering and motion segmentation show the superiority of the proposed method. |
