Matrix completion via capped nuclear norm

Matrix completion is to recover a low-rank matrix from a subset of its entries. One of the solution strategies is based on nuclear norm minimisation. However, since the nuclear norm is defined as the sum of all singular values, each of which is treated equally, the rank function may not be well approximated in practice. To overcome this drawback, this study presents a matrix completion method based on capped nuclear norm (MC-CNN). The capped nuclear norm can reflect the rank function more directly and accurately than the nuclear norm, Schatten p-norm (to the power p) and truncated nuclear norm. The relation between the capped nuclear norm and the truncated nuclear norm is revealed for the first time. Difference of convex functions’ programming is employed to solve MC-CNN. In the proposed algorithm, a key sub-problem, i.e. a matrix completion problem with linear regularisation term, is solved by using the active subspace selection method. In addition, the algorithm convergence is discussed. Experimental results show encouraging results of the proposed algorithm in comparison with the state-of-the-art matrix completion methods on both synthetic and real visual datasets.