Research On Nonlinear Sparse Tangent Subspace Clustering Method

With the outbreak of the big data in the information age,a large number of data need to be dealt with in all walks of life.They are not only large scale,high dimension but also have complex structure.High dimensional data generally have more complex nonlinear structure.However,the high-dimensional data with manifold structure can be mapped to each low dimensional subspace,which can be seen as reduction and clustering simultaneously.Manifold learning is one of the commonly method of dimensionality reduction,which is assumed that a low dimensional data manifold embedded in high-dimensional space.In other words,the essence of data is low dimensional.Then low dimensional manifold data can be found by the distance relationship between data and the purpose of dimension reduction will be realized.Although manifold learning can find out the geometric structure of the data,it can only be applied to a single data manifold.In the more general case,the structure of high-dimensional data is interleaved with multiple sub-manifolds of different dimensions,and different manifolds come from different low-dimensional subspaces.Subspace clustering is designed to find out the data manifolds from different subspaces to achieve clustering of multiple data manifolds.This method is widely used in image processing(image compression,image representation),computer vision problems(image Segmentation,motion segmentation,etc.),system identification,machine learning and other fields.Spectral clustering is one of the basic methods of subspace clustering.Sparse subspace clustering and low rank subspace clustering are all based on it.Sparse subspace clustering find out a sparse expression of data in the data space that means the coefficients represented by the same subspace are as non-zero as possible,and the coefficients represented by other subspaces are as zero as possible.The idea of low-rank subspace clustering is similar to the former,and the difference is that the coefficient matrix of the linear representation data is restricted by the low rank because each sparse vector in the matrix can't deduce the low rank of the matrix.Sparse subspace clustering and low rank subspace clustering are only applicable to the data manifolds in linear subspaces.In order to generalize linear subspaces to nonlinear subspaces,a clustering method is proposed as sparse spheroid space clustering(STSC).This method constructs a convex optimization problem of weighted ?1 norm.The weighted coefficients contain the local and global geometric structure information of the nonlinear data manifold,which have a advantage that the subspaceof the nonlinear data can be obtained by using linear expression.In this paper,we propose a weighted gradient operator to obtain the closed solution of the norm approximation under the constraint condition and use the Augmented Lagrangian multiplier(ALM)for the weighted ?1 norm in the proposed convex optimization problem to obtain the sparse expression of the whole convex optimization problem.Finally,the STSC algorithm has been applied on three typical clustering problems such as simple data set,motion segmentation and face clustering in this paper.At the same time,several common subspace clustering methods are simulated on the same data set and we analyze and compare their advantages and disadvantages.