Non-linear matrix completion.

•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.