Unbiased risk estimates for singular value thresholding and spectral estimators

Emmanuel J. Candes, Carlos A. Sing-Long, Joshua D. Trzasko

Research output: Contribution to journalArticlepeer-review

115 Scopus citations


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.

Original languageEnglish (US)
Article number6545395
Pages (from-to)4643-4657
Number of pages15
JournalIEEE Transactions on Signal Processing
Issue number19
StatePublished - 2013


  • 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


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