  In an increasing number of applications, it is of interest to recover an
approximately low-rank data matrix from noisy observations. This paper 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 which applies a soft-thresholding rule to the singular
values of the noisy observations. Among other things, our formulas offer a
principled and automated way of selecting regularization parameters in a
variety of problems. In particular, we demonstrate the utility of the unbiased
risk estimation for SVT-based denoising of real clinical cardiac MRI series
data. We also give new results concerning the differentiability of certain
matrix-valued functions.
