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We present a method for supervised learning of sparsity-promotingregularizers, a key ingredient in many modern signal reconstruction problems.The parameters of the regularizer are learned to minimize the mean squarederror of reconstruction on a training set of ground truth signal andmeasurement pairs. Training involves solving a challenging bilevel optimizationproblem with a nonsmooth lower-level objective. We derive an expression for thegradient of the training loss using the implicit closed-form solution of thelower-level variational problem given by its dual problem, and provide anaccompanying gradient descent algorithm (dubbed BLORC) to minimize the loss.Our experiments on simple natural images and for denoising 1D signals show thatthe proposed method can learn meaningful operators and the analytical gradientscalculated are faster than standard automatic differentiation methods. Whilethe approach we present is applied to denoising, we believe that it can beadapted to a wide-variety of inverse problems with linear measurement models,thus giving it applicability in a wide range of scenarios.

Bilevel learning of l1-regularizers with closed-form gradients(BLORC)