Novel Tensor Decompositions And High-Performance Algorithms With Applications To High-Dimensional Data Restoration

The rapid development of science and technology has given rise to the wide pres-ence of high-dimensional data,such as video data,remote sensing images,and traffic flow data.However,due to many unavoidable reasons,the high-dimensional data inevitably suffers from noise interference or elements missing.High-dimensional data restoration aims to recover high-quality data from its degraded observation,which is one of the hot and difficult issues in the current cross-research of mathematics and information.Mathe-matically,the high-dimensional data restoration problem can be modeled as an ill-posed inverse problem.Therefore,prior knowledge of high-dimensional data needs to be in-troduced to stabilize the solution process.In recent years,thanks to the powerful abil-ity to capture the global correlation of high-dimensional data,tensor decomposition has achieved great success in high-dimensional data restoration problems.However,there are still many deficiencies in the existing tensor decomposition methods.This dissertation aims at the key mathematical problems in high-dimensional data restoration,focusing on a fundamental problem:developing new tensor decomposition methods and tensor ranks,and two practical problems:establishing tensor optimization models for high-dimensional data restoration problems and designing high-performance solving algorithms.The main research contents,contributions,and innovations are listed below.Firstly,the omnidirectional tensor singular value decomposition is proposed,which solves the problem that the traditional tensor singular value decomposition cannot flexibly describe different correlations of different dimensions in tensors.Based on omnidirec-tional tensor singular value decomposition,a new tensor rank,called tensor fibered rank,is defined.Since directly minimizing the sum of elements of the tensor fibered rank is a non-deterministic polynomial hard problem,its convex relaxed form and non-convex relaxed form are proposed.Based on the relaxed forms,two mixed noise removal models for hyperspectral images are established,and the corresponding solving algorithms based on the alternating direction method of multipliers are designed.Extensive numerical ex-periments demonstrate that the proposed method outperforms compared methods in noise removal and intrinsic structure preservation of hyperspectral images.Secondly,based on the first part,a new tensor unfolding operator is defined,which unfolds an Nth-order tensor into a third-order tensor by reordering its mode-k₁,k₂slices lexicographically.Then,a new tensor rank,called tensor N-tubal rank,is defined,which is suitable for Nth（N≥3）-order tensors and can flexibly describe different correlations of different dimensions in tensors.To effectively minimize the sum of the elements of the N-tubal rank,its convex relaxation form is defined.Then,the corresponding high-dimensional data completion and sparse noise removal models are established and the alternating direction method of multipliers-based solving algorithms are designed.Exten-sive numerical experiments demonstrate that the proposed method outperforms compared methods on completion and denoising problems.Thirdly,a novel fully-connected tensor network（FCTN）decomposition is proposed,which decomposes an Nth-order tensor into a series of low-dimensional Nth-order fac-tors and establishes a multi-linear operation between any two factors.The FCTN de-composition has two advantages:one is that it can fully describe the correlation between any two dimensions of tensors?the other is that it has transposition invariance in math-ematics.Based on the FCTN decomposition,the FCTN rank is further proposed.To verify the superiority of the FCTN decomposition,it is applied to the high-dimensional data completion problem and a corresponding tensor optimization model is established.A solving algorithm based on the proximal alternating minimization method is designed and the theoretical convergence of the algorithm is proved.Numerical experimental re-sults demonstrate that the FCTN decomposition-based method is superior to other tensor decomposition-based methods.Fourthly,based on the third part,a novel tensor completion model is proposed by introducing a factor-based regularization to the framework of the FCTN decomposition,which can simultaneously characterize the global low-rankness and local continuity of tensors.A solving algorithm based on the proximal alternating minimization method is designed and the theoretical convergence of the algorithm is proved.According to the structural characteristics of subproblems in the solving algorithm,a fast and accurate so-lution strategy is designed.Extensive numerical experimental results show that the pro-posed method is superior to other methods in the restoration of the global structure and local detail information of high-dimensional images.