Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators
E. J. Candès, C. A. SingLong, and J. D. Trzasko

In an increasing number of applications, it is of interest to recover an approximately
lowrank 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).
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 wellmodeled by lowrank 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.

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