Research On Theory And Applications Of Low-Tubal-Rank Tensor Recovery

With the rapid development of modern information technologies,such as big data and artificial intelligence,traditional sparse recovery methods represented by compressed sensing and low-rank matrix recovery have been unable to deal with the large-scale,high-dimensional and complex tensor data encountered in many fields,such as psychological measurement,medical imaging,computer vision,pattern recognition,etc.So it is urgent to establish a set of new models,theory and methods of information processing according to tensor data.Therefore,by utilizing the hypothesis of the low rank property of tensors,the low-rank tensor recovery technology for the purpose of dealing with some practical application problems in big data environment came into being.However,the effectiveness of its application strongly depends on a thorough understanding of tensor data,such as the establishment of effective and computable decomposition strategies,the definition of simple and reasonable rank functions,etc.Based on the recently developed Tensor Singular Value Decomposition(T-SVD)framework and the corresponding definition of tensor tubal rank,in this dissertation,we explore two typical problems of low-tubal-rank tensor recovery,namely high-order compressed sensing and tensor robust principal component analysis.The main work can be summarized into four aspects as follows:Firstly,considering that the measurement signal may be disturbed by additive noise and the design of algorithm,we propose a Regularized Tensor Nuclear Norm Minimization(RTNNM)model for low-tubal-rank tensor recovery.By introducing the notions of T-SVD and tensor tubal rank,we initiatively define a novel Tensor Restricted Isometry Property(TRIP)which is a natural extension of the restricted isometry property in compressed sensing and low-rank matrix recovery on tensors.With the help of the crucial analysis tool of the T-RIP,the mathematical condition of robust recovery of low-rank tensor is established by solving the RTNNM model,and the error upper bound estimate of the robust recovery is obtained.The relationship between regularization parameters and recovery accuracy in the RTNNM model is analyzed by numerical experiments,which provides a reference for the algorithm design in practical applications.Secondly,because the obtained sufficient conditions of robust recovery of low-tubal-rank tensor require that the linear measurement maps satisfy the T-RIP,by developing and using the high-dimensional probability tools,such as concentration inequalities,?-net,covering numbers,etc.,we prove that under suitable conditions on the number of measurements,if the elements of the random measurement ensemble follow a generalized sub-Gaussian distribution,including zero-mean Gaussian distributions,symmetric Bernoulli distributions and all zeromean bounded distributions,it will satisfy the T-RIP with high probability.The obtained minimal possible number of linear measurements is very reasonable and order optimal from the perspective of the degrees of freedom of a tensor.Moreover,the validity of the required number of measurements is verified by numerical experiments.Thirdly,we study the most essential tensor tubal rank minimization model in low-tubalrank tensor recovery,and examine the fundamental question of the minimal number of linear observations needed to recover a tensor from these observations,regardless of the practicality of the reconstruction scheme.Taking advantage of linear algebra,differential manifolds,set theory,etc.,the uniqueness guarantee of the solution of the tensor tubal rank minimization model is established,that is,we provide the estimation of the minimum number of samples required for uniform recovery and non-uniform recovery using tensor tubal rank minimization method.These two benchmark results are of great theoretical significance to judge whether the cost of sampling number of different alternative recovery methods,including tensor nuclear norm minimization method,is reasonable or not.Fourthly,although Tensor Principal Component Pursuit(TPCP)is a powerful approach in tensor robust principal component analysis,TPCP needs to meet some relatively strict tensor incoherence conditions and the accuracy of tensor recovery also needs to be improved.In view of this,we consider the row and column space knowledge of some clean data in the tensor data stream as the subspace prior information of the low-tubal-rank tensor to be recovered,and propose the modified tensor incoherence conditions and recovery model,which incorporates the prior subspace information.In addition,we use the tools of probability concentration inequalities and dual certificates to establish the corresponding theoretical results of the model.It is proved that the prior subspace information can not only weaken the tensor incoherence assumptions,but also significantly improve the recovery quality of tensor data.The promising performance of the proposed method is supported by simulations and real data applications including color video recovery and face image recovery.