Journal of the
Korean Mathematical Society JKMS

The matrix completion problem is to predict missing entries of a data matrix using the low-rank approximation of the observed entries. Typical approaches to matrix completion problem often rely on thresholding the singular values of the data matrix. However, these approaches have some limitations. In particular, a discontinuity is present near the thresholding value, and the thresholding value must be manually selected. To overcome these difficulties, we propose a shrinkage and thresholding function that smoothly thresholds the singular values to obtain more accurate and robust estimation of the data matrix. Furthermore, the proposed function is differentiable so that the thresholding values can be adaptively calculated during the iterations using Stein unbiased risk estimate. The experimental results demonstrate that the proposed algorithm yields a more accurate estimation with a faster execution than other matrix completion algorithms in image inpainting problems.

Keywords: Matrix completion, singular value thresholding, Stein unbiased risk estimator

MSC numbers: Primary 15A18, 15A23

Supported by: This work was supported by Hankuk University of Foreign Studies Research Fund and National Research Foundation of Korea (NRF) grant funded by the Korean government (2018R1C1B5085022).