Research On Sparse Subspace Clustering Algorithm And Theory Under Noise

High dimensional data is widespread in the fields of machine learning,image processing,pattern recognition and so on.In high dimensional space,However,it is a common phenomenon that the distance between data is almost equal everywhere,Which has formed a great challenge to the traditional clustering method using the distance among datasets as a measure of the similarity.As an effective algorithm to solve the problem of high dimensional data clustering,the subspace clustering algorithm attracts the researchers'attention.Recently,the subspace clustering methods based on sparse representation?SR?and low-rank?LRR?have been increasing interest in many fields.In this paper,we study the subspace clustering algorithm based on sparse representation and low rank representation,by the research and analysis,this paper proposes the sparse subspace recovery theory of noisy data,and subspace clustering method based on TL₁ norm Constraints.The main contents are as follows:First of all,the study of sparse subspace coefficients recovery theory.This section considers the problem of recovering sparse subspace coefficients under noise.A constrained₁l minimization is considered and study conditions under which the solution of the proposed optimization satisfies the robust sparse subspace recovery condition.More specifically,we show that the coefficients corresponding to data points from the same subspaces with a small error,or the l₂-norm of the coefficients corresponding to the data points in other subspace will be sufficiently small.Secondly,research on subspace clustering algorithm.The TL₁ norm is applied to propose a new optimization model for the study of subspace clustering.Although the optimization is nonconvex,in the case of non-noise,it proves that the optimal solution of the proposed model is the coefficient matrix with block-diagonal structure,which provides the theoretical guarantee for the latter spectral clustering.In the case of dealing with noise,the constraint condition of this model is presented to be equivalent with the optimal model using the corrected data as the dictionary,which contributes to improving the clustering accuracy.Then,the alternating direction method of Lagrangian multipliers is applied to solving the unknown matrices.Experimental results show that subspace clustering method based on TL₁ norm not only enhances the sparsity of coefficient matrix,but also is superior to low-rank subspace clustering and sparse subspace clustering method in terms of clustering accuracy and robustness to noise.