High-dimensional Data Separation And Reconstruction Via(non) Convex Minimization

With the advent of the information age,we often confront with a variety of complex and valuable high-dimensional data in production and daily life.How to effectively mine and process these high-dimensional data has triggered a heated discussion in academia and industry.Compressed sensing is a novel and effective high-dimensional data processing theory,which can exactly reconstruct a signal with high probability by utilizing the sparsity and compressibility of signal data.Currently,Compressed sensing is being extensively applied in various fields of science and engineering,including compressive imaging,medical imaging,pattern recognition,image processing,etc.Based on the theory of compressed sensing and the application background,this paper makes a study on the different types of high-dimensional data processing.The main works of this thesis are as follows:Chapter one outlines the background and significance of the study of compressed sensing theory,and briefly introduces the latest research progress of compressed sensing and the application results.Chapter two particularly presents the three major aspects of compressed sensing that are the sparse representation of signal,the design of measurement matrix,and the reconstruction theories and algorithms.Chapter three firstly introduces the compressed data separation model for multi-modal data,and secondly,investigates perturbations of compressed data separation with redundant tight frames via non-convex D-?q-minimization.By exploiting the properties of the redundant tight frames and the perturbation matrix,i.e.,mutual coherence,null space property and restricted isometry property,the condition on reconstruction of sparse signal with redundant tight frames is established and the error estimation between the local optimal solution and the original signal is provided.The obtained results show that D-?q-minimization is robust and stable for the reconstruction of sparse signal with redundant tight frames.Chapter four mainly investigates the signals which are block-sparse under redundant tight frames based on convex ?2/?1-minimization method and Block D-RIP theory.Under the condition 0 < ?2k|?< 0.2,the obtained results show that ?2/?1-minimization method can robustly reconstruct the original signal,meanwhile,improve the existing reconstruction condition and error upper bound.Using the discrete Fourier transform dictionary,we conducted a series of simulation experiments which sufficiently verified the theoretical results.Chapter five investigates the low-rank tensor completion problem.Based on the fact that singular values before the target rank does not affect rank minimization of tensors,we propose low-rank tensor completion via partial sum minimization of singular values algorithm(PSSV-LRTC).Some experiments are performed on both synthetic data and real applications,all results show that our algorithm has a higher precision and convergence rate than previous work.Chapter six is summary and prospect of the further valuable research on perturbations of compressed data separation,block-sparse compressed sensing and low-rank tensor completion.