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A generalized nonconvex algorithm framework for low-rank and sparse matrix decomposition
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DOI：10.1007/s10489-025-06971-8.png)
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Abstract
En 中文
The low-rank and sparse matrix decomposition problem is a hot and challenging problem in computer science. In this paper, we consider it as a nonconvex relaxation optimization problem by using a family of nonconvex functions to approximate the rank function and the $$\ell _{0}$$ -norm in low-rank and sparse matrix decomposition problem, namely, generalized low-rank and sparse matrix decomposition problem. The essence of this paper is to develop an adaptive algorithm framework with parameters updating for the nonconvex relaxation problem. Firstly, we prove the equivalence between the generalized low-rank and sparse matrix decomposition problem and the regularization generalized low-rank and sparse matrix decomposition problem. This means that the optimal solution of generalized low-rank and sparse matrix decomposition problem can be exactly obtained by solving its regularization minimization problem. Secondly, we present a tractable nonconvex algorithm framework to solve the regularization generalized low-rank and sparse matrix decomposition problem. The convergence analysis of the algorithm framework is provided. More importantly, we also define a very powerful parameter-setting strategy to adapt the optimal parameters in iteration of the proposed algorithm framework. Finally, we test the proposed algorithms on some random low-rank and sparse matrix decomposition problems, and the numerical results verified the effectiveness of the proposed algorithms. In addition, we also extend the proposed algorithms to the image denoising and background modeling from surveillance video.
Keywords:
Low-rank and sparse matrix decomposition
Nonconvex function
regularization Reneralized low-rank and sparse matrix decomposition problem
Nonconvex algorithm framework
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