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 that 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.
- Differentiability of eigenvalues and eigenvectors
- Stein's unbiased risk estimate (SURE)
- magnetic resonance cardiac imaging
- singular value thresholding
ASJC Scopus subject areas
- Signal Processing
- Electrical and Electronic Engineering