Proximal Splitting Derivatives for Risk Estimation

Charles Deledalle (1) , Samuel Vaiter (1) , Gabriel Peyré (1) , Jalal M. Fadili (2) , Charles H Dossal (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 Equipe Image - Laboratoire GREYC - UMR6072
3 IMB - Institut de Mathématiques de Bordeaux

This paper develops a novel framework to compute a projected Generalized Stein Unbiased Risk Estimator (GSURE) for a wide class of sparsely regularized solutions of inverse problems. This class includes arbitrary convex data fidelities with both analysis and synthesis mixed L1-L2 norms. The GSURE necessitates to compute the (weak) derivative of a solution w.r.t.~the observations. However, as the solution is not available in analytical form but rather through iterative schemes such as proximal splitting, we propose to iteratively compute the GSURE by differentiating the sequence of iterates. This provides us with a sequence of differential mappings, which, hopefully, converge to the desired derivative and allows to compute the GSURE. We illustrate this approach on total variation regularization with Gaussian noise and to sparse regularization with poisson noise, to automatically select the regularization parameter.

Keywords: Sparsity; regularization; inverse problems; risk estimator; GSURE; automatic differentiation

Domains: Signal and Image Processing; Signal and Image processing

Charles Deledalle, Samuel Vaiter, Gabriel Peyré, Jalal M. Fadili, Charles H Dossal. Proximal Splitting Derivatives for Risk Estimation. NCMIP'12, Apr 2012, France. pp.012003, ⟨10.1088/1742-6596/386/1/012003⟩. ⟨hal-00670213⟩