# A Bilevel Optimization Approach for Parameter Learning in Variational Models

Karl Kunisch^{*} and Thomas Pock

The problem of parameter learning for variational image denoising models is address. The learning problem is formulated as a bilevel optimization problem, where the lower level problem is given by the variational model and the higher level problem is expressed by means of a loss function that penalizes errors between the solution of the lower level problem and the ground truth data. A class of image denoising models incorporating $\ell_p$-norm based analysis priors using a fixed set of linear operators. The resulting non-smooth bilevel optimization problems are solved by semi-smooth Newton methods and it is shown that the optimized image denoising models can achieve state-of-the-art performance.

Mathematics Subject Classification: 49J52 49N45 68U10

Keywords: Regularization parameter; image denoising; learning theory; non-differentiable optimization; bilevel optimization; semi-smooth Newton algorithm

Minisymposion: Noise Estimation, Model Selection and Bilevel Optimization