Weighted Power Factorization for Robust Matrix Completion

‍ ‍Matrix completion was a problem of filling up the missing elements of a partially observed matrix. It had attracted a lot of research interests in many fields， including image inpainting and recommendation systems. After the introduction of the nuclear norm heuristic theory， various methods were proposed to solve matrix completion problems. A line of works utilizing singular value decomposition （SVD） worked well but suffered from the high computational complexity of SVD. To tackle this problem， the power factorization （PF） model was proposed and applied in matrix completion problems. PF-based methods factorized the matrix into the product of two rank-r matrices in advance， which could also be viewed as factorizing the matrix into the sum of r rank-1 components. In this way， the low-rank characteristic of the matrix was naturally satisfied. Thus， the computationally expensive SVD operations could be avoided. However， the PF-based model had to estimate a relatively exact rank parameter r， which was realistically difficult. Besides， it assigned equal weights to those rank-1 components， which might limit the performance of PF. In this paper， to address the aforementioned problems， we proposed a weighted power factorization （WPF） model for matrix completion problems. Introduced an auxiliary weight variable with sparsity constraint whose role was twofold. First， it could capture the importance and assign different weights to those rank-1 components. Second， it could locate and drop out the unnecessary rank-1 components. This approach could overcome the limitations of the PF-based models and improve the performance of the matrix completion problem. Furthermore， considered the circumstances where entities observed of the matrix are contaminated with non-Gaussian noises. In this circumstance， we combined the information theory and utilized correntropy to formulate the WPF model for robust matrix completion. First transformed the formulation of WPF model with half-quadratic （HQ） theory and then optimized it through alternating gradient descent （AGD）. Revealed the automatic rank selection advantage of our proposed WPF method and demonstrated its superior performance compared with SVD-based and PF-based methods under noisy conditions through experiments on synthetic data and real image data. Moreover， explored the wider applications of WPF method in image inpainting.