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 com- pletion, 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 ap- proximation, 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 it- erations 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. 相似文献
The morphologies of triblock copolymer/homopolymer blend films confined between two neutral hard walls were studied via MC simulations on a simple cubic lattice. For ABA/A and ABA/B blend films, the effects of φh (the volume fraction of the homopolymer) and Mh/Mb (the ratio of the molecular mass of the homopolymer to that of the corresponding blocks) on the morphologies were investigated in detail. For both ABA/A and ABA/B blend films, a higher φh or Mh/Mb would result in stronger macrophase separation between the triblock copolymer and homopolymer. For ABA/C blend films, Mh/Mb hardly influences the morphologies of homopolymer domains regardless of whether the homopolymer C is more compatible with block A or with block B. Compared to AB/A and AB/C blend films, the morphologies of ABA/A (or ABA/B) and ABA/C blend films are much more irregular. The simulated results in this work show good consistency with experiments and other simulations.