On The Degrees Of Freedom in Total Variation Minimization

Feng Xue; Thierry Blu

ICASSP 2020accepted0 citations

On The Degrees Of Freedom in Total Variation Minimization

Abstract

In the theory of linear models, the degrees of freedom (DOF) of an estimator play a pivotal role in risk estimation, as it quantifies the complexity of a statistical modeling procedure. Considering the total-variation (TV) regularization, we present a theoretical study of the DOF in Stein’s unbiased risk estimate (SURE), under a very mild assumption. First, from the duality perspective, we give an analytic expression of the exact TV solution, with identification of its support. The closed-form expression of the DOF is derived based on the Karush-Kuhn-Tucker (KKT) conditions. It is also shown that the DOF is upper bounded by the nullity of a sub-analysis-matrix. The theoretical analysis is finally validated by the numerical tests on image recovery.

BibTeX

@inproceedings{icassp2020_onthedegreesoffr,
  title = {On The Degrees Of Freedom in Total Variation Minimization},
  author = {Feng Xue and Thierry Blu},
  booktitle = {ICASSP 2020},
  year = {2020}
}