Research On Tensor Completion And Denoising Algorithms Based On Tensor Factorization

With the development of computer technology,the Internet,and the Internet of Things,etc,the data generated by human society is becoming more and more complex and the dimensions are getting higher and higher,which puts higher demands on data analysis.At the same time,many data in the real world often have data loss and noise.How to deal with such data has become one of the important issues in the fields of machine learning,data mining,and computer vision,etc.To solve the problems of data loss and noise in multi-dimensional data,based on tensor decomposition and optimization theory,this dissertation focuses on tensor completion and robust tensor decomposition.The main research contents are as follows:1.An ST-HOSVD based method is proposed for the problem of tensor completion.Firstly,since most tensor-completion models based on tensor decomposition take the rank of the tensor as a hyperparameter,researchers offen achieve the low rank of the final recovery results by giving a small value of the rank in advance.How to select the appropriate rank is a difficult problem to deal with.To this end,based on ST-HOSVD,a fast adaptive tensor decomposition algorithm for rank selection is proposed in this dissertation.Based on it,the "completiondecomposition" iterative process is used to complete the tensor with missing values.In the decomposition process,a number of low-rank approximation of the original tensor are constructed,and the recovery result is obtained by the average of them,with the accuracy further improved.2.A mixing noise denoising algorithm based on tensor completion is proposed for the problem of Gaussian noise and impulsive noise in the image.Since the image data is usually smooth in the local part,the median filter is used to detect the non-smooth pixel points in the image as the possible position of the impulse noise,which are discarded afterwards.Then the image recovery process is treated as a tensor completion problem.The proposed method considers the effect of Gaussian noise while removing impulse noise.In order to reduce the influence of sharp details in the image during the denoising process,the information of the noisy pixels is used again here to regenerate a new restored image,which balances the contradiction between denoising and retaining image details.Further,the case of noise patches is also considered herein.In response to this problem,the denoising algorithm of the above discrete case is generalized,and the block noise is "corroded" step by step through the iterative process until it completely disappears.3.A generalized Tucker model is proposed to improve robustness to impulse noise.For the robust decomposition problem of tensors containing impulse noise,Tucker based on L2 norm is not competent.In view of the fact that when p<2,the Lp norm can effectively reduce the sensitivity of the model to outliers,the Tuckerp model is proposed,which extends the traditional Tucker decomposition model based on the Lp norm constraint to solve the problem.The construction method is used to approximate the solution of the problem with a series of Tucker decomposition submodels.At the same time,it is proved that the proposed method is convergent under certain condictions,and when it converges,its solution is the KKT solution of the Tuckerp model,and its second-order necessary condition is also satisfied under certain conditions.4.A robust tensor orthogonal decomposition model based on Huber loss function is proposed for simultaneous data loss and impulse noise pollution.In the process of model solving,the construction method is used to construct a series of simple tensor complement submodels to approximate the solution of the proposed model.At the same time,the theoretical analysis proves that the proposed method is convergent,and the KKT solution of the model can be achieved when it converges.In addition,using the characteristics of Huber loss function,an equivalent form of the proposed model and relevant theoretical analysis are given.