Research On Robust Sparse Recovery Algorithms And Applications Based On Information Theory

In the current era of big data,data are often characterized by high dimension,large capacity,diversification and rapid growth.It not only means the increase of useful in-formation in the data,but also can bring a huge burden to data storage,transmission and computation.Nowdays,how to effectively mine and utilize the low-dimensional struc-ture of data to process high-dimensional data has been an important challenge faced by various industries.In recent years,the rise of robust sparse recovery research indicates that the sparsity of data can be utilized to analyze the high-dimensional data effectively.And when certain conditions are met,the original high-dimensional data can be accurately reconstructed from the limited observations.As a theory of information acquisition and transmission,information theory shows us a novel way to explore the problem of robust sparse recovery.Guided by information theory,this dissertation takes numerical opti-mization and variational Bayes as the main mathematical tools,comprehensively utilizes the basic principles and relevant measures of information theory,and combines with com-pressed sensing image recovery,near-field millimeter wave imaging,multispectral imag-ing and other specific applications to conduct in-depth research on robust sparse recovery,the main contributions of this dissertation include the following four parts:First,by means of the the q-Gaussian function obtained from maximizing Tsallis en-tropy,a robust information theory measure,namely q-Gaussian generalized correntropy is deirved.We propose a robust formulation for sparse signal reconstruction from com-pressed measurements corrupted by impulsive noise,which exploits the q-Gaussian gen-eralized correntropy(1<q<3)as the loss function for the residual error and utilizes a?₀-norm penalty term for sparsity inducing.To solve this formulation efficiently,we develop a gradient based adaptive algorithm which incorporates a zero-attracting regu-larization term into the framework of adaptive filtering.This new proposed algorithm blending the advantages of adaptive filtering and q-Gaussian generalized correntropy can obtain accurate reconstruction and satisfactory robustness with a proper shape parameter q.Numerical experiments identify that the proposed algorithm is able to get better recov-ery results compared with some mainstream robust sparse recovery algorithms under the same impulsive noise conditions.Second,a novel formulation which combines the M-estimator and the non-convex regularization term is presented to address the issue of robust sparse recovery in impulsive noise.As a robust measure,M-estimator is not only closely related to the correntropy,but also shows its strong ability of suppresssing impulsive noise in many applications.Meanwhile,the non-convex regularization is capable of overcoming the biased estimation problem induced by the convex?₁-norm regularization and thus can obtain more accurate reconstruction results.Furthermore,to solve the resulting non-convex formulation,an efficient low computing complexity algorithm is derived by means of half-quadratic(HQ)optimization.The complexity of proposed algorithm is also analyzed.Reconstruction experiments of simulated sparse signals and real images in impulsive noise demonstrate the effectiveness of the proposed algorithm.Third,guided by the principle of information theory,a robust sparse recovery Bayesian method based on generalized approximate message passing is proposed.We comprehen-sively utilize the relative entropy,channel coding and belief propagation in the derivation of this method.The proposed method can be seen as an effective combination of infor-mation theory and variational Bayes method.By constructing the compressive sampling imaging model and corresponding sensing matrix,the proposed method is applied to near-field millimeter wave imaging and the real-time imaging is realized.Experimental results confirm that the algorithm can rapidly and effectively reconstruct 2-D millimeter wave images from under-sampled measurements.Finally,the object of our research is extended from one-dimensional signal and two-dimensional image to multi-dimensional tensor data.And the robust tensor recovery prob-lem in the environment of impulsive noise is studied.A novel robust tensor recovery for-mulation based on correntropy and hybrid tensor sparsity measure is proposed.We analyze the reason why the proposed robust tensor recovery formulation can effectively suppress the influence of impulsive noise and the advantage of the hybrid tensor sparsity measure.Besides,to solve this formulation effectively,an efficient large-scale optimization algo-rithm is derived based on the framework of alternating direction method of multipliers(ADMM).The results of multispectral imaging recovery indicate that the proposed algo-rithm can obtain robust tensor recovery in the environment of impulsive noise.