Projected gradient descent based on soft thresholding in matrix completion

Abstract

Matrix completion is the extension of compressed sensing. In compressed sensing, we solve the underdetermined equations using sparsity prior of the unknown signals. However, in matrix completion, we solve the underdetermined equations based on sparsity prior in singular values set of the unknown matrix, which also calls low-rank prior of the unknown matrix. This paper firstly introduces basic concept of matrix completion, analyses the matrix suitably used in matrix completion, and shows that such matrix should satisfy two conditions: low rank and incoherence property. Then the paper provides three reconstruction algorithms commonly used in matrix completion: singular value thresholding algorithm, singular value projection, and atomic decomposition for minimum rank approximation, puts forward their shortcoming to know the rank of original matrix. The Projected Gradient Descent based on Soft Thresholding (STPGD), proposed in this paper predicts the rank of unknown matrix using soft thresholding, and iteratives based on projected gradient descent, thus it could estimate the rank of unknown matrix exactly with low computational complexity, this is verified by numerical experiments. We also analyze the convergence and computational complexity of the STPGD algorithm, point out this algorithm is guaranteed to converge, and analyse the number of iterations needed to reach reconstruction error. Compared the computational complexity of the STPGD algorithm to other algorithms, we draw the conclusion that the STPGD algorithm not only reduces the computational complexity, but also improves the precision of the reconstruction solution.