Fast and robust rank-one matrix completion via maximum correntropy criterion and half-quadratic optimization

Matrix completion refers to seeking a low-rank matrix to match an incomplete matrix and fill its missing entries, and is an important topic because many real-world data can be modeled as low-rank matrices. Most existing schemes rely on ℓ₂-norm minimization and hence are not robust to outliers. Even the ℓ₁-norm based robust matrix completion algorithms proposed in the literature have poor recovery performance for large gross errors, and require knowing the matrix rank, which is difficult to accurately determine in practice. In this paper, we devise a robust and fast rank-one matrix completion algorithm via combining the maximum correntropy criterion (MCC) and half-quadratic (HQ) optimization theory. The MCC, i.e., minimizing the Welsch cost function, can resist the gross errors but it is non-convex. While HQ optimization can transform the Welsch cost function into a quadratic form, making the resultant optimization problem easy to solve. Furthermore, an adaptive kernel width strategy is derived and there are no tunable parameters except for the termination conditions in our algorithm. Computer simulations using synthetic data and experimental results of real-world images demonstrate that the developed method achieves accurate recovery performance and high computational efficiency.