Low-rank matrix completion is an important theme both theoretically and practically. However, the state-of-the-art methods based on convex optimization usually lead to a certain amount of deviation from the original matrix. To perfectly recover a data matrix from a sampling of its entries, we consider a non-convex alternative to approximate the matrix rank. In particular, we minimize a matrix γ-norm under a set of linear constraints. Accordingly, we derive a shrinkage operator, which is nearly unbiased in comparison with the well-known soft shrinkage operator. Furthermore, we devise two algorithms, non-convex soft imputation (NCSI) and non-convex alternative direction method of multipliers (NCADMM), to fulfil the numerical estimation. Experimental results show that these algorithms outperform existing matrix completion methods in accuracy. Moreover, the NCADMM is as efficient as the current state-of-the-art algorithms.
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