Low-Rank Block Matrix Learning Under RKHS

Many problems in practical applications can be summarized as a matrix factorization problem.In the big data environment,many machine learning content can be expressed in the form of a matrix,such as image recognition and other visual fields.However,in practical applications,hundreds of data are often involved.The complexity of traditional matrix-based data processing technology will increase quadratically as the scale of the problem increases,which makes many applications impossible.In-spired by sparse,low-rank and block techniques such as kernel methods,subspace clustering and low-rank representation,the following work is proposed to solve the above problems:Firstly,the original data is mapped to the regenerated kernel Hilbert space(RKHS)through the kernel method,which can solve the nonlinear problem of the subspace data,thereby effectively learning the original data characteristics.The kernel method can effectively deal with complex nonlinear problems.Through the kernel function,the non-linear data that is difficult to process is mapped to the high-dimensional feature space and converted into linear data that is easy to process.The kernel method has the following two advantages:(1)The kernel function used in the kernel method is designed for practical applications,which is conducive to integrating the prior knowledge of the data.(2)Kernel method maps nonlinear data into high-dimensional space by using kernel techniques,therefore,the high-dimensional space does not increase the difficulty of the iterative processing of the algorithm.This is very helpful for processing data sets that have a wide range of sources and many features in practical applications,which cannot be clustered in low-dimensional spaces.Secondly,in order to make the distance between different classes as large as pos-sible and the distance between data in the same class as small as possible,improved discrimination information is introduced to restrict it.Unlike traditional discriminant information,traditional discriminant information usually uses distance-based algorithms or variance to constrain the intra-class and inter-class relationships in the data space.The improved discrimination information mainly restricts the intra-class and inter-class relationship of the data in the coefficient space.Compared with constraints in a huge and complex data space,constraints in the coefficient space are simpler and more effective.Thirdly,drawing on the idea of neighbors,a constraint on the second-order neigh-bors of the coefficient matrix is proposed.In simple terms,second-order neighbors means that if a data point is a second-order neighbor of another data point,first they must be neighbors,and secondly they must have at least one common neighbor.Only by satisfying the above two points can it be called a second-order neighbors.Second-order neighbors can optimize the sparsity and connectivity of the subspace,thereby effectively solving the problem of data overlap in the edge region of the subspace and improving the efficiency of subspace clustering.Finally,we use subspace clustering technology to cluster.Subspace clustering can effectively cluster high-dimensional data,mainly through local search of related dimensions.In the high-dimensional data space,traditional clustering algorithms are difficult to process data,mainly because:(1)There is a lot of noise in high-dimensional space,and it is very difficult to cluster the data set in it.(2)Compared with low-dimensional space,the distribution of data in high-dimensional space is more sparse,and the distance between data is close to the same.Most clustering algorithms are calculated based on the distance between data,which is difficult to apply to data in high-dimensional space.The core of the subspace clustering algorithm is to use the self-expression property of the subspace,that is,each point in the subspace can be represented by a linear combination of other points in the same subspace.This can effectively process data in high-dimensional space.In summary,combining the research content of the above four parts,a low-rank block matrix learning method under RKHS is proposed,which can effectively process data with a wide range of sources and complex features in current practical applications and effectively cluster it.