Research On Algorithm And Application For Low-Rank High-Order Tensor Recovery Based On Singular Value Decomposition

With the emerging information technologies,such as big data,artificial intelligence,evolving at a rapid pace,the tensor data characterized by large volume,high dimension and complex multi-factor structure have been increasing in the application fields of computer vision,pattern analysis and so on.Generally,these tensor data are easily affected by various degradation factors in the process of acquisition and transmission,like noise pollution,information missing,outliers interference.Recently,the low rank tensor recovery has become a popular method for processing and analyzing the degraded tensor data.However,existing methods based on tensor Singular Value Decomposition(t-SVD)mainly focus on practical problems related to third-order tensors,while order-d(d≥ 4)tensors are commonly encountered in real-world applications,like color videos,traffic flow dataset,and bidirectional texture functions images dataset.In addition,the algebraic theory induced by the order-3 t-SVD has taken shape,while the one based on the order-d t-SVD is relatively insufficient,which greatly restricts its wide application in Low-Rank High-Order Tensor Recovery(LRHOTR).Therefore,it is particularly critical and urgent to develop the model,algorithm and theory for LRHOTR by establishing the order-d t-SVD algebraic framework,especially for the typical and extendable Low Rank High-Order Tensor Completion(LRHOTC).This paper first proposes a generalized order-d t-SVD algebraic framework,and then makes a deep study on the LRHOTC problem.The main research results are listed below:Aimed at high-order tensor data with missing values,an efficient and scalable LRHOTC model induced by the Generalized Tensor Nuclear Norm(GTNN)is devised within the proposed order-d t-SVD algebraic framework.We theoretically prove that under certain order-d tensor incoherence conditions,the established model can achieve exact completion for any order-d low t-SVD rank tensors with missing values with an overwhelming probability.Further,the Alternating Direction Method of Multipliers(ADMM)framework is utilized to design an efficient solution algorithm,and verify the validity of the proposed model and the correctness of the established theory on the synthetic tensor data,then apply it to a series of real-world applications,e.g.,face inpainting,hyperspectral videos completion,color videos restoration and light-field images recovery.Emperical studies on synthetic data and real-world visual data illustrate that compared with other state-of-the-art recovery algorithms,the proposed one achieves highly competitive performance in terms of both qualitative and quantitative metrics.Aimed at high-order tensor data that contains missing values and noises/outliers simultaneously,we investigate a flexible and robust LRHOTC model,in which the low-rank component is constrained by the Generalized Weighted Tensor Schanttenp(0<p≤1)Norm that approximates the t-SVD rank of high-order tensors more tightly than the GTNN,while the sparse component is regularized by its lq(0<q≤1)norm.Then,based on the generalized soft threshold strategy and the order-d rt-SVD integrated with acceleration idea,an efficient ADMM solution algorithm with convergence guarantee is designed to solve the formulated double non-convex model,through which the simultaneous denoising and completion for arbitrary low t-SVD rank high-order tensors can be achieved.Experiments on both synthetic and real-world data show that our method is capable of reconstructing the low-rank structure of tensors with the better accuracy and robustness against several state-of-the-art methods.