Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning

• URL: http://arxiv.org/abs/2203.02189v1
• Date: Fri, 4 Mar 2022 08:50:50 GMT
• Title: Matrix Completion via Non-Convex Relaxation and Adaptive Correlation Learning
• Authors: Xuelong Li, Hongyuan Zhang, Rui Zhang
• Abstract summary: We develop a novel surrogate that can be optimized by closed-form solutions. We exploit upperwise correlation for completion, and thus an adaptive correlation learning model.
• Score: 90.8576971748142
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The existing matrix completion methods focus on optimizing the relaxation of rank function such as nuclear norm, Schatten-p norm, etc. They usually need many iterations to converge. Moreover, only the low-rank property of matrices is utilized in most existing models and several methods that incorporate other knowledge are quite time-consuming in practice. To address these issues, we propose a novel non-convex surrogate that can be optimized by closed-form solutions, such that it empirically converges within dozens of iterations. Besides, the optimization is parameter-free and the convergence is proved. Compared with the relaxation of rank, the surrogate is motivated by optimizing an upper-bound of rank. We theoretically validate that it is equivalent to the existing matrix completion models. Besides the low-rank assumption, we intend to exploit the column-wise correlation for matrix completion, and thus an adaptive correlation learning, which is scaling-invariant, is developed. More importantly, after incorporating the correlation learning, the model can be still solved by closed-form solutions such that it still converges fast. Experiments show the effectiveness of the non-convex surrogate and adaptive correlation learning.

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