Adaptive thresholding for iterative matrix completion with heterogeneous missing probability: H-AdaptiveImpute

Abstract

Several low-rank matrix completion algorithms have been developed, which often employ thresholded singular value decomposition (SVD) iteratively. Their performance depends on the choice of thresholding tuning parameters and initial input matrix. In particular, the initialization of the non-convex optimization algorithms plays a pivotal role. However, they often use a zero matrix or a consistent low-rank matrix estimator under the simple homogeneous observation probability assumption as the initial input matrix. This paper proposes a non-convex low-rank matrix completion algorithm with an intelligent initial matrix under the heterogeneous observation probability structure. Specifically, the proposed algorithm iterates the thresholded SVD starting from a tailored initialization based on the estimator of the singular values and vectors of the underlying matrix. We show that the proposed initial estimator is consistent, and the adaptive thresholding tuning parameters are estimated using theoretically justified and data-driven procedures. The results of the numerical experiments show that the proposed algorithm performs well compared to the existing method.