Study On Matrix Rank Minimization Algorithms And Applications Based On Non-convex Approximation

Due to the advance of the technology of sensor, communication and computer, human beings could get massive data at all times. However, these data are often large scale, high dimensional and contain different types of noises, which brings difficulties and challenges such as data storage,transmission and analysis. Therefore, it is a hot topic to reduce dimensionality and denoise in the field of information science, medicine and life science. The key technology of compressed sensing, the matrix rank minimization technique, has attracted more attention in the fields of high dimensional data analysis, image processing and computer vision. This thesis focuses on the nonconvex approximation of matrix rank minimization and its applications in computer vision and hyperspectral image denoising. Thesis's main work is as follows:First, solving the nuclear norm-based minimization problem usually leads to suboptimal solution of the original rank minimization problem. To address this issue, I propose a nonconvex surrogate based on log-determinant function to approximate the matrix rank function. An iterative algorithm based on the augmented Lagrangian multipliers method is developed to solve the nonconvex optimization problem. Under ?>1 , we prove that our proposed algorithm is convergent to a stationary point of the original nonconvex problem. Finally, the algorithm is applied to the multi task learning and subspace clustering experiments to verify the effectiveness of our proposed algorithm.Then, for the general matrix rank minimization model in which the original data is mixed with different kinds of noises, including robust principal component analysis and low rank presentation problems, nonconvex approximation matrix rank minimization model based on Laplace function is presented. I adopt a linearization technique and develop an iterative algorithm based on the augmented Lagrangian multipliers method. The empirical studies for practical applications demonstrate that the non-convex regularizer is a tighter approximation to the original matrix rank function rather than the traditional nuclear norm and our proposed algorithm outperforms many other state-of-the-art convex and nonconvex methods.Next, for hyperspectral image denoising problem, I apply the nonconvex low rank matrix decomposition to linearly decompose a degraded hyperspectral image into a low rank matrix representing the clean image, a sparse matrix denoting non-Gaussian noise and an unstructured matrix estimating Gaussian noise. In this process, the linear function between the matrix rank function and the l2,1 norm of the sparse matrix is taken as the objective function, subjecting to the linear decomposition of a matrix. Then, the augmented Lagrangian multiplier method is used to solve the problem, consequently, I could get a new iterative algorithm. Numerical results are presented to illustrate the effectiveness of the proposed algorithm by comparing it with the existing state-of-the-art low rank matrix recovery algorithms.Finally, the main content of this paper is summarized and the further research project is proposed based on some shortcomings of the existing algorithms.