Non-linear matrix completion. (May 2018)

• Record Type: Journal Article
• Authors: Fan, Jicong; Chow, Tommy W.S.
• Is Part Of: Pattern recognition. Volume 77(2018:May)
• Page Start: 378
• Page End: 394
• Publication Date: 2018-05
• Subjects: Matrix completion -- Low-rank -- Kernel -- Schatten p-norm -- Image inpainting -- Single-/multi-label classification -- Non-linear denoising; Pattern perception -- Periodicals; Perception des structures -- Périodiques; Patroonherkenning; 006.4
• DOI: 10.1016/j.patcog.2017.10.014
• Languages: English
• ISSNs: 0031-3203

Abstract

Highlights: Conventional matrix completion methods are linear methods. This paper proposed a non-linear matrix completion (NLMC) method that is able to handle data of non-linear structures and high-rank matrices. NLMC significantly outperforms existing methods in the tasks of image inpainting and single-/multi-label classification. The idea of NLMC is extended to a non-linear rank-minimization framework applicable to other problems such as non-linear denoising. Abstract: Conventional matrix completion methods are generally linear because they assume that the given data are from linear transformations of lower-dimensional latent subspace and the matrix is of low-rank. Therefore, these methods are not effective in recovering incomplete matrices when the data are from non-linear transformations of lower-dimensional latent subspace. Matrices consisting of such nonlinear data are always of high-rank or even full-rank. In this paper, a novel method, called non-linear matrix completion (NLMC), is proposed to recover missing entries of data matrices with non-linear structures. NLMC minimizes the rank (approximated by Schatten p -norm) of a matrix in the feature space given by a non-linear mapping of the data (input) space, where kernel trick is used to avoid carrying out the unknown non-linear mapping explicitly. The proposed NLMC is compared with existing methods on a toy example of matrix completion and real problems including image inpainting and single-/multi-label classification. The experimental results verify the effectiveness and superiority of the proposed method. In addition, the idea of NLMC can be extended to a non-linear rank-minimization framework applicable to other problems such as non-linear denoising.