Research On Tensor Low-Rank Modeling And Algorithm For High-Dimensional Data Restoration Problems

With the advance of data collection equipment,the acquired data shows a trend of high dimensionality,providing more feature information in subsequent analysis.However,affected by factors such as acquisition equipment or acquisition environment,the acquired high-dimensional data often have degradations.The high-dimensional data recovery problem aims to recover the underlying data from the degraded data,which is an ill-posed inverse problem,and its stable and effective solution depends on the prior knowledge of the underlying data.Since high-dimensional data are usually highly correlated in all dimensions,low-rank is an important prior knowledge for high-dimensional data restoration.In this dissertation,we focus on two key issues in low-rank modeling:the accurate characterization of the intrinsic low-rank structure of high-dimensional data,and the compact and faithful low-rank representation of high-dimensional data.To address these issues,two novel low-rank regularizers and two novel subspace representations are designed,and the corresponding low-rank tensor optimization model and efficient solving algorithm are proposed to solve the real-world high-dimensional data restoration problem.The research contents and innovations are summarized as follows:Firstly,a tensor ε pseudo-norm is proposed to solve the problem of poor Tucker rank approximation by existing convex relaxations.The proposed ε pseudo-norm achieves an approximate unbiased estimation of the Tucker rank by measuring the singular values using a non-convex function,which guarantees sparsity promotion and is solvable at the same time.Compared with the matrix norm,the ε pseudo-norm can adequately characterize the low-rankness of all tensor modes.Based on the ε pseudo-norm,a mixed noise removal model for hyperspectral images is established,and a solving algorithm based on the augmented Lagrange multiplier method is designed.Numerical experimental results show that the proposed method outperforms the matrix norm-based method and the Tucker decomposition-based method for the reconstruction of hyperspectral images.Secondly,the kernelized tensor nuclear norm is proposed to achieve implicit lowrank characterization of high-dimensional data.For nonlinear high-dimensional data with high-rank characteristics,the kernelized tensor nuclear norm applies a nonlinear mapping to the transformed frontal slices to capture their implicit low-rankness.In the computation of the proposed norm,the computation in the high-dimensional latent space is transformed into the inner product in the low-dimensional original space by introducing the kernel method,which reduces the computational complexity.Based on the kernelized tensor nuclear norm,a tensor robust kernel principal component analysis model is established and a solving algorithm based on the alternating direction multiplier method is designed,and the proposed method can effectively decompose the observed data into the implicit low-rank component and the sparse component.The numerical experimental results verify that the proposed method is superior to the robust principal component analysis method based on tensor nuclear norm.Thirdly,the coupled matrix subspace representation is proposed to exploit the latent correlation of multi-temporal remote sensing images(MTRSIs).In the matrix subspace representation of the single-temporal image,the basis matrix represents the spectral features of the mixed endmembers and the coefficients represent the abundance of the mixed endmembers.By coupling the coefficients of subspace representations corresponding to different time nodes,a coupled matrix subspace representation is proposed to compactly characterize the consistency of the abundance and the variability of the spectral features of the MTRSIs at different times.Based on the coupled matrix subspace representation,a thick cloud removal model for MTRSIs is established,and a solving algorithm based on the augmented Lagrange multiplier method is designed and analyzed for its theoretical convergence.The numerical experimental results show that the proposed method can achieve a better thick cloud removal effect with lower time cost.Fourthly,the tensor subspace representation is proposed to achieve a more faithful and compact low-rank representation of the tensor.Based on the tensor singular value decomposition framework,the t-linear representation and tensor subspace representation are defined.The tensor subspace basis has a stronger representation ability than the matrix subspace basis for high-dimensional data.Based on the tensor subspace representation,the nonlocal self-similarity of the coefficient tensor is inscribed,a Gaussian noise removal model for hyperspectral images is established,and a solution algorithm based on the proximal alternating minimization method is designed and proved to be theoretically convergent.The numerical experimental results verify that the proposed tensor subspace-based method is superior to the matrix subspace-based method.