Matrix Decomposition Based Low-rank Tensor Recovery Algorithm And Its Application

The expression of data is one of the most fundamental problems for data analysis and pro-cessing.Vector and matrix are the major representation forms.The real data is often affected by a variety of factors,which leads to the high dimension and the multiple structure of the data obtained.Vector and matrix are limited in the representation of the complicated structure,while tensor can be an alternate for it is a high-order generation of vector and matrix.Tensor can do better in preserving and describing the inner structure of data,especially the multilinear structure.Therefore,tensor is the main topic in this thesis.The data can be lost,polluted or mixed with others during the process of collecting,storing and transferring.Recovering the original data from their degraded observation is a classic inverse problem,it cannot be solved without prior knowledge of the original data as constrains.Inspired by the achievements about theories,algorithms and applications in the field of compressed sens-ing and low-rank matrix recovery,the low-rank tensor recovery problem attracts more and more interest.At present,the theory and algorithm for low-rank tensor recovery are focused on how to describe the rank of tensor and how to utilize the low-rank property in the process of calculation.For low-rank matrix recovery,numerous algorithms are based on the relaxation of matrix rank by nuclear norm,because in this case,nuclear norm is the best convex approximation of matrix rank.While for tensor,this conclusion is not true in theory,so the accuracy of the algorithms based on the tensor nuclear norm cannot be compared with that in matrix case,which may affect the application of low-rank tensor recovery.Therefor,for the low-rank tensor recovery problem,there still need discussion about the model,algorithm and application.In this thesis,I first describe the research background,significance,development of low-rank tensor recovery,and its connection to sparse vector and low-rank matrix recovery.Then study its algorithms and applications in three subproblems:completion,approximation and ro-bust principal component analysis.For the representation and analysis of the low-rank part in the model and algorithm,unlike the traditional idea of nuclear norm relaxation,in this thesis low-rank matrix decomposition is adopted for all mode matricization of the underlying low-rank tensor.Based on the alternating direction strategy,iterating algorithms are designed for the(Augmented)Lagrangian of the opti-mal models.In the iteration,two rank adjustment strategies are proposed.After the convergence analysis of the algorithm,the numerical experiments of synthetic data are also designed.The experimental results show the effectiveness of the proposed model and algorithm.The proposed algorithm is also applied to video completion,multiframe image denoising and video background modeling as the applications of those three subproblems.The experimental results for these real-world data show that the new algorithm proposed in this paper can be a new choice in related problems.