On The Degrees Of Freedom In Total Variation Minimization

In the theory of linear model, the degrees of freedom (DOF) of an estimator plays a pivotal role in the risk estimation, as it quantifies the complexity of a statistical modeling procedure. Considering the total-variation (TV) regularization, we in this paper present a theoretical study of the DOF in the Stein’s un- biased risk estimate (SURE), under a very mild assumption. First, from the duality perspective, we give an analytic expression of the 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.