Decomposition Approach for Low-Rank Matrix Completion and Its Applications

Abstract

In this paper, we describe a low-rank matrix completion method based on matrix decomposition. An incomplete matrix is decomposed into sub-matrices which are filled with a proposed trimming step and then are recombined to form a low-rank completed matrix. The divide-and-conquer approach can significantly reduce computation complexity and storage requirement. Moreover, the proposed decomposition method can be naturally incorporated into any existing matrix completion methods to attain further gain. Unlike most existing approaches, the proposed method is not based on norm minimization nor on SVD decomposition. This makes it possible to be applied beyond real domain and can be used in arbitrary fields, including finite fields. The effectiveness of our proposed method is demonstrated through extensive numerical results on randomly generated and real matrix completion problems and a concrete application-video denoising. The numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed significantly faster than recent algorithms. In the proposed denoising approach, we present a patch-based video denoising algorithm by grouping similar patches and then formulating the problem of removing noise using a decomposition approach for low-rank matrix completion. Experiments show that the proposed approach robustly removes mixed noise such as impulsive noise, Poisson noise, and Gaussian noise from any natural noisy video. Moreover, our approach outperforms state-of-the-art denoising techniques such as VBM3D and 3DWTF in terms of both time and quality. Our technique also achieves significant improvement over time against other matrix completion methods.