Adaptive shrinkage of singular values

To recover a low-rank structure from a noisy matrix, truncated singular value decomposition has been extensively used and studied. Recent studies suggested that the signal can be better estimated by shrinking the singular values as well. We pursue this line of research and propose a new estimator offering a continuum of thresholding and shrinking functions. To avoid an unstable and costly cross-validation search, we propose new rules to select two thresholding and shrinking parameters from the data. In particular we propose a generalized Stein unbiased risk estimation criterion that does not require knowledge of the variance of the noise and that is computationally fast. A Monte Carlo simulation reveals that our estimator outperforms the tested methods in terms of mean squared error on both low-rank and general signal matrices across different signal-to-noise ratio regimes. In addition, it accurately estimates the rank of the signal when it is detectable.