Research On Signal Recovery Method Based On Sparse And Low Rank

In modern data and signal processing,signal estimation is the main task of restoring objects of interest.However,as the dimensions of signals and data keep increasing,the limitations of hardware and storage resources make the need for low sample complexity signal recovery approaches more and more urgent.This thesis is devoted to understanding the underlying low-dimensional features of high-dimensional signals,and exploiting the low dimensional structural information of signals to design low sample complexity signal recovery approaches.In this thesis,the variational Bayesian and non-convex optimization are the main mathematical tools.This thesis focuses on joint sparse and low rank matrix recovery and phase recovery,which are two very related sub-problems of low rank matrix recovery.First,we focus on the channel estimation problem in millimeter-wave broadband wireless communication,and study the signal processing problem of reconstructing joint sparse and low rank matrix from linear compressed observation data.Channel estimation in millimeter-wave wireless communication is a very challenging problem.In recent years,in the measurement and modeling of millimeter-wave channel it has been found that the millimeter-wave channel not only has sparse scattering characteristics,but also has the characteristics of joint sparseness and low rank.By exploiting this joint sparse and low rank structures,the pilot overhead of the millimeter-wave channel estimation can be further reduced,and spectral efficiency can be improved.This thesis proposes a novel joint sparse and low rank inducing prior model and then designs a new joint sparse and low rank matrix recovery method via variational Bayesian learning.The joint sparse and low rank features of the matrix allow accurate estimation of millimeter wave signals with less pilot overhead and without the rank prior of the channel matrix.The simulation results show that compared with the existing channel estimation methods,the joint sparse and low rank Bayesian learning method proposed in this paper has lower sampling complexity and stronger anti-noise ability.Secondly,the low rank phase retrieval problem is studied,where the phase and amplitude of the low rank structure signal need to be recovered from the amplitude measurement of multiple observations.Due to the temporal correlation of the signals,when the observation data of a plurality of sampling moments are jointly processed,the signal matrix to be estimated has a low rank characteristic.Based on the inherent low rank feature of the signal,a Gaussian mixture probabilistic prior model that promotes low rank is introduced.Based on this,a low rank phase retrieval method via variational Bayesian learning is proposed.Compared with the existing nonconvex two-stage method based on alternating minimization,the low rank phase retrieval via variational Bayesian learning proposed in this thesis has lower sample complexity and is more robust to initialization point selection.The third part of the thesis studies the robust initialization algorithm of the nonconvex phase retrieval and the fast nonconvex two-stage phase retrieval algorithm.Find an initial point closer to the optimal solution's convex neighborhood is the key to the success of the nonconvex phase retrieval method.In this thesis,we use the null space and subspace information of the signal to design the efficient and robust initialization estimators.Three null vector initialization methods are proposed,which can provide a robust and high quality initial point for the nonconvex phase retrieval algorithms.Compared with the existing null vector initialization method,spectral initialization algorithms and its variants,the orthogonal decomposition based initialization algorithms introduced in this thesis have significant performance advantages and superior ability to resist additive noise,which can further enhance the performance of existing nonconvex phase retrieval algorithms.In addition,inspired by the theory of escaping saddle points theory,we propose a fast non-convex phase retrieval algorithm via the momentum acceleration strategy.The experimental results show that the performance of the nonconvex phase retrieval algorithm can be further enhanced by adopting this momentum acceleration strategy with the ability to escape from the saddle points.