Low-rank and sparse matrices fitting algorithm for low-rank representation

• Published in: Computers & mathematics with applications (1987) Vol. 79; no. 2; pp. 407 - 425
• Main Authors: Zhao, Jianxi, Zhao, Lina
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 15.01.2020; Elsevier BV
• Subjects: Accuracy; Acoustics; Algorithms; Alternating direction minimization; Clustering; Computational geometry; Computer simulation; Convergence; Convexity; Decomposition; Iterative methods; Linear search method; Low-rank representation; Low-rank subspace; Newton method; Noise reduction; Object recognition; Optimization; Pattern recognition; Representations; Sparse matrices; Sparsity; Speech recognition; Speech sounds; Subspaces; Walking; Low-rank subspace; Low-rank representation; Newton method; Alternating direction minimization; Linear search method
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/j.camwa.2019.07.012

In real world, especially in the field of pattern recognition, a matrix formed from images, visions, speech sounds or so forth under certain conditions usually subjects to a low-rank subspace structure. Sparse noise, small noise and so on can be eliminated by the low-rank property of this matrix, leading to the well-known low-rank representation problem. At present, existing algorithms for this problem still need to be improved in the aspects of the recovery accuracy of low-rank component and sparse component, the clustering accuracy of subspaces’ data and their convergence rate. This paper proposes a low-rank matrix decomposition non-convex optimization extended model without nuclear norm. Motivated by human walking, we combine the direction and step size iterative formula with the alternating direction minimization idea for the sake of decomposing the original optimization model that is difficult to be solved into three comparatively easily solved sub-optimization models. On the basis of these, Low-Rank and Sparse Matrices Fitting Algorithm (LSMF) is presented for the sub-models in this paper, which quickly alternates the search direction matrices and the corresponding step sizes. Theoretically, it is proved that LSMF converges to a stable point of the extended model. In simulation experiments, better results are achieved in the three aspects under appropriate conditions. The face denoising and background/foreground separation further demonstrate the capability of LSMF on handling large-scale and contaminated dataset.