Nonconvex Regularized Methods For Sparse Optimization And Applications

With the advent of the era of big data,the data we are facing also have blowout growth.Traditional methods are quite difficult to analyze large scale data.Now s-parse optimization provides an effective way,and it has been widely used in the fields of signal and image processing,machine learning,pattern recognition and so on.In this paper,by introducing nonconvex regularized optimization model,we study sig-nal recovery,image reconstruction,foreground extraction for video surveillance,and establish corresponding nonconvex optimization model,theory and algorithm.In signal recovery,we consider the l1-lp minimization problem.In theory,we derive computable lower bounds for nonzero entries of the generalized first-order sta-tionary points of l1-lp minimization,and hence of its local minimizers.In algorithms,based on three locally Lipschitz continuous ?-approximation to lp norm,we design several iterative reweighted l1 and l2 methods to solve those approximation problems.Furthermore,we show that any accumulation point of the sequence generated by these methods is a generalized first-order stationary pointof ll-lp minimization.This result,applies to the iterative reweighted l1 methods based on the new Lipschitz continuous?-approximation,provided that the approximation parameter ? is below a threshold val-ue.Numerical results are also reported to demonstrate the efficiency of the proposed methods.In image reconstruction,we propose a novel nonconvex total variation(TV)model for image deblurring and denoising,which combines a nonconvex data fitting term and a nonconvex regularization term perfectly.The alternating direction method of multipliers(ADMM)is proposed to solve the noncovnex TV optimization problem.The resulting subproblems either have closed-form solutions or can be solved by fast solvers,which makes the ADMM particularly efficient.In theory,with the help of the smoothing technique and Kurdyka-Lojasiewicz function,we prove that the sequence generated by the ADMM converges to a stationary point when the penalty parameter is above a computable threshold.Extensive numerical experiments illustrate that our proposed nonconvex TV model outperforms the existing convex and nonconvex TV models.In particular,for the image with a high percentage of extreme-value pixels,the increase is 24.14dB.In foreground extraction,we study a nonconvex TV and structured sparsity regu-larized RPCA model for video surveillance with dynamic background.The TV is used for noise reduction of the foreground,and structured sparsity is used to characterize the overall structure of the background.In order to solve the model,we propose a sym-metric Gauss-Seidel method based on ADMM,and establish its convergence analysis.We also calculate some numerical experiments to demonstrate its performance.In ad-dition,we extend it to tensor based nonconvex RPCA model,which can make better use of the spatio-temporal characteristics.In fact,to the best of our knowledge,this is the first time to integrate the nonconvex penalty into a tensor framework,which has demonstrated the superiority of performance.In summary,we consider signal recovery,image reconstruction,foreground extrac-tion,propose nonconvex regularized optimization models,design efficient algorithms,give convergence analysis,and verify the efficiency through extensive numerical exper-iments.