Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators

In an increasing number of applications, it is of interest to recover an approximately low-rank data matrix from noisy observations. This work develops an unbiased risk estimate  —  holding in a Gaussian model  —  for any spectral estimator obeying some mild regularity assumptions. In particular, we give an unbiased risk estimate formula for singular value thresholding (SVT), a popular estimation strategy. These formulas may help in offering a principled and automated way of selecting regularization parameters in a variety of problems; for instance, for the denoising of real clinical cardiac MRI series data (see the paper).

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The left figure shows the main ideas behind our method in an MRI application. In a cardiac MR sequence, the structures appearing in most image blocks do not show major changes: these individual sequences are well-modeled by low-rank matrices, and a good estimate can be obtained by using SVT. The unbiased estimate provides an accurate approximation of the MSE, thus allowing us to choose the optimal parameter for the SVT estimate.