Selecting thresholding and shrinking parameters with generalized SURE for low rank matrix estimation by Julie Josse, Sylvain Sardy

To estimate a low rank matrix from noisy observations, truncated singular value decomposition has been extensively used and studied: empirical singular values are hard thresholded and empirical singular vectors remain untouched. Recent estimators not only truncate but also shrink the singular values. In the same vein, we propose a continuum of thresholding and shrinking functions that encompasses hard and soft thresholding. To avoid an unstable and costly cross-validation search of their thresholding and shrinking parameters, we propose new rules to select these two regularization 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. In addition, it automatically selects the rank of the matrix. A Monte Carlo simulation reveals that our estimator outperforms the tested methods in terms of mean squared error and rank estimation.

The attendant R implementation of the algorithm is on Julie Josses's site.

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