  We present a method for supervised learning of sparsity-promoting
regularizers for denoising signals and images. Sparsity-promoting
regularization is a key ingredient in solving modern signal reconstruction
problems; however, the operators underlying these regularizers are usually
either designed by hand or learned from data in an unsupervised way. The recent
success of supervised learning (mainly convolutional neural networks) in
solving image reconstruction problems suggests that it could be a fruitful
approach to designing regularizers. Towards this end, we propose to denoise
signals using a variational formulation with a parametric, sparsity-promoting
regularizer, where the parameters of the regularizer are learned to minimize
the mean squared error of reconstructions on a training set of ground truth
image and measurement pairs. Training involves solving a challenging bilievel
optimization problem; we derive an expression for the gradient of the training
loss using the closed-form solution of the denoising problem and provide an
accompanying gradient descent algorithm to minimize it. Our experiments with
structured 1D signals and natural images show that the proposed method can
learn an operator that outperforms well-known regularizers (total variation,
DCT-sparsity, and unsupervised dictionary learning) and collaborative filtering
for denoising. While the approach we present is specific to denoising, we
believe that it could be adapted to the larger class of inverse problems with
linear measurement models, giving it applicability in a wide range of signal
reconstruction settings.
