A Study On The Hybird Algorithms Of Matrix Recovery Regularization And Application

This dissertation is a study on regularization algorithm of ma-trix recovery,improving the traditional convex optimization algorithm(NNM).NNM is easy to solve and has been proved to be able to obtain an exact so-lution under some conditions on the linear transformation d.However,the nuclear norm minimization requires more measurements for exact recov-ery of the low rank solution,and we cannot achieve a stable solution from the Lasso model when the predictors are highly correlated or the number of predictors to recover is much bigger than the number of predictors.In this dissertation,we mainly study l2-l1 regularization algorithms for matrix recovery based on WNNM and Schatten p-norm individually.We also study the Schatten p-norm minimization algorithm for image denois-ing.The main contents are as follows:1.We propose a hybrid regularization model(also called elastic-net)of matrix recovery based on weighted nuclear norm minimization,aimed at taking into account the sparsity and stability simultaneously.We also pro-pose a fast and efficient algorithm(MH-WNNM)to solve the model.We prove its convergence by an upper bound under the constraints on the loss functions,which in theory ensures that MH-WNNM can gets arbitrarily close to a limit point.Experimental results show that the proposed algo-rithm can obtain a more stable solution than the Lasso model.Meanwhile,we obtain more accurate results than the traditional convex optimization algorithm.2.In order to improve the defect of the nuclear norm minimization which requires more measurements for exact recovery of low rank solu-tion,we propose a new matrix elastic-net regularization based on Schatten p-norm minimization(MEN-Sp)which can also achieve a stable solution.In order to derive the solution of this nonconvex model,two optimization algorithms are presented,including a new alternate iterating algorithm and MEN-Sp algorithm.Numerical experiments demonstrate that MEN-Sp al-gorithm can produce exact reconstruction with fewer measurements.3.We propose an image denoising model based on nonlocal self-similarity(NSS)method.By stacking nonlocal similar patches into a ma-trix,we then add the Schatten p-norm of the matrix to the model.In order to solve the model,we change the model formally and propose the corre-sponding iterative algorithm.The experimental results show that the pro-posed algorithm can effectively recover the original matrix,and can lead to visible PSNR improvements over state-of-the-art methods.