Adaptive weighting function for weighted nuclear norm based matrix/tensor completion

Weighted nuclear norm provides a simple yet powerful tool to characterize the intrinsic low-rank structure of a matrix, and has been successfully applied to the matrix completion problem. However, in previous studies, the weighting functions to calculate the weights are fixed beforehand, and do not change during the whole iterative process. Such predefined weighting functions may not be able to precisely characterize the complicated structure underlying the observed data matrix, especially in the dynamic estimation process, and thus limits its performance. To address this issue, we propose a strategy of adaptive weighting function, for low-rank matrix/tensor completion. Specifically, we first parameterize the weighting function as a simple yet flexible neural network, that can approximate a wide range of monotonic decreasing functions. Then we propose an effective strategy, by virtue of the bi-level optimization technique, to adapt the weighting function, and incorporate this strategy to the alternating direction method of multipliers for solving low-rank matrix and tensor completion problems. Our empirical studies on a series of synthetic and real data have verified the effectiveness of the proposed approach, as compared with representative low-rank matrix and tensor completion methods.