Research And Application Of Principal Component Analysis Algorithm For Low-rank Tensor Decomposition

Principal component analysis plays an important role in computer vision applications and data bioinformatics.However,traditional principal component analysis is easily disturbed by severely damaged or abnormal observations,and these phenomena are ubiquitous in the real world,so the robust principal component analysis guarantees a certain degree of anti-interference ability.In the real world,many data,such as color images,videos,and 4-D f MRI data,are multi-dimensional,and simple matrixization will destroy the internal structure of the data.Tensor decomposition is a powerful calculation tool,which can naturally describe high-dimensional data.Naturally,by modeling and separating the low-rank and sparse components in the data,the tensor robust principal component analysis is proposed,and the extension of various matrix optimization methods to tensors has become a recent research hotspot.The principal component analysis method of tensor proposes different models according to different tensor decomposition.Classical tensor decomposition methods include CANDECOMP/PARAFAC(CP)decomposition and Tucker decomposition.In recent years,decomposition models such as chain decomposition and ring decomposition of tensors have been proposed,and there are different applications based on these tensor models.In this paper,the robust principal component analysis based on tensor chain and tensor ring decomposition has been studied.The main research work includes the following two aspects.A non-convex tensor principal component analysis method(referred to as the TTlog TRPCA algorithm)is introduced to separate tensors by modeling the target tensor with low-rank properties to achieve the purpose of removing noise.This method is based on tensor chain decomposition.A new non-convex tensor robust principal component analysis model is proposed.The model aims to extract low-rank tensor components damaged by sparse noise.In order to solve the model,alternate iterations are used.Multiplier method for optimization iteration.Numerical experiments can show the effectiveness and convergence of the TT-log TRPCA algorithm,and its performance is not inferior to similar algorithms.Introduced the theoretical guarantee and decomposition method of tensor ring decomposition,inspired by traditional tensor principal component analysis,naturally proposed a tensor robust principal component analysis model of balanced tensor ring expansion.The TRNN method is proposed based on the convex rank function approximation of the nuclear norm,and the logTRNN method is proposed based on the non-convex rank function approximation of the logarithmic function,and the update format of specific variables is given.Then analyze the complexity and convergence of the two methods.Finally,it can be seen from numerical experiments that tensor ring decomposition has superior denoising performance in recovering color images with higher noise ratios,black and white videos and color videos.In particular,the logTRNN method based on non-convex optimization far exceeds other methods.Finally,The influence of different parameters on the logTRNN algorithm is discussed.