Unbiased Risk Estimates for Singular Value Thresholding and Spectral Estimators
E. J. Candès, C. A. Sing-Long, and J. D. Trzasko
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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).
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.
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