Research On High Dimension Data Recovery Method Based On Tensor Decomposition

In the era of big data and artificial intelligence,more and more decisions need to be made based on massive data analysis,which often requires the collection of a large amount of information.A notable feature of this information is that it often has a complex structure(or multiple labels)and a higher dimension.Matrix,as a method for processing two-dimensional data,often ignores the relation within the data when it is used to process high-dimensional data.The existing literature shows that the tensor-based model can better mine the internal structural information of data.This makes tensors widely used in machine learning,pattern recognition,signal processing,data mining and other fields.However,due to many practical reasons,tensor data often face the problems of data loss and noise pollution.Therefore,how to use the incomplete data to recover or estimate the original data is very important.This paper focuses on the design and application of tensor completion method,which consists of Chapter 3 and Chapter 4.Tensor(Matrix)nuclear norms have been successfully used to promote the lowrankness of tensors(Matrices)in low-rank tensor completion.However,singular value decomposition(SVD),which is computationally expensive for large-scale matrices,frequently appears in solving those nuclear norm minimization models.Based on the tensortensor product(T-product),in Chapter 3,we first establish the equivalence between the so-called transformed tubal nuclear norm for a third-order tensor and the minimum of the sum of two factor tensors’ squared Frobenius norms under a general invertible linear transform.Gainfully,we introduce a mode-unfolding(often named as “spatio-temporal”in the internet tra c data recovery literature)regularized tensor completion model that is able to e ciently exploit the hidden structures of tensors.Then,we propose an algorithm using alternating direction multiplier method to solve the proposed tensor optimization model,and obtain its convergence and computational complexity analysis.It is remarkable that our approach does not require any SVDs and all subproblems of our algorithm enjoy closed-form solutions.A series of numerical experiments on tra c data recovery,color images and videos inpainting demonstrate that our SVD-free approach takes less computing time to achieve satisfactory accuracy than some state-of-the-art tensor nuclear norm minimization approaches.In Chapter 4,we propose a tensor completion model based on the third-order tensor train(TT)decomposition.In this model,the sparse regularization and the “spatiotemporal” regularization are introduced to characterize the sparsity of the kernel tensor and the inherent block similarity of the data,respectively.According to the structural characteristics of the problem,some auxiliary variables are introduced to convert the original model into a separable form equivalently,and the method of combining proximal alternating minimization(PAM)and alternating direction multiplier method(ADMM)is used to solve the model.Numerical experiments show that the introduction of two regular terms is beneficial to improve the stability and practical effect of data recovery,and the proposed method is superior to other methods.When the sampling rate is low or the image is structurally missing,the presented method is more effective.