Fast Tensor Principal Component Extraction And Tracking

Robust principal component analysis separates the original signal into a low-rank component and a sparse component.The low-rank component can be used to analyze the principal component of the signal or dimensionality reduction of the high-dimensional signal.Sparse component can be used to remove noise from data or to detect objects of interest.This general assumption makes it widely used in signal processing,data analysis,machine learning and other fields.Tensor Robust Principal Component Analysis（TRPCA）is its extension form in multidimensional signal field.TRPCA methods can be divided into online methods and offline methods according to sequential relation of data processing.The principal component analysis of tensors under two modes is studied in this thesis.This thesis mainly studies TRPCA problems in two modes.This thesis mainly studies the TRPCA problems under two modes.For the TRPCA problem in offline mode,the existing methods mainly solve this problem from the perspective of model designing or mathematical optimization,but ignore the possible prior information inside the tensor data,which limits the generalization performance of the algorithm on the real data.In this thesis,Fourier transform in tensor singular value decomposition is used to solve the TRPCA problem from the perspective of frequency filtering in signal processing.In particular,this thesis decompose the tensors to be processed into the sum of some frequency components,and then design the filtering coefficients of each frequency band based on the prior information obtained from frequency component analysis of the data.By introducing frequency-domain filtering into each iteration of the TRPCA problem,the frequency-filtered TRPCA method（Frequencyfiltered TRPCA,FTRPCA）is proposed.Experimental results show that this way of using prior information extracts the principal components of tensors efficiently and shows the superiority in various metrics in various application scenarios.For the online TRPCA problem,the existing methods learns a stable low-rank subspace,and transform the online TRPCA problem into the projection problem of sequential samples in the subspace.However,such methods usually cannot quickly capture the real subspace when the subspace changing with time.In this thesis,a dynamic subspace tracking algorithm is proposed,which introduces a moving window to make the model focus on the samples in the window and eliminate the influence of the samples outside the window on the model.Based on this algorithm,a dynamic TRPCA（DTRPCA）algorithm is proposed.Numerical experiments which simulates various scenarios show the robustness of the DTRPCA algorithm to subspace changes,and it also proves the effectiveness of the DTRPCA algorithm.Then in the real-time video tracking task scenario,the experimental results show that the DTRPCA algorithm is capable of dynamically capturing the changing scene,and the results of the algorithm are significantly better than the online algorithm of the same type.