Learning sparsity-promoting regularizers for linear inverse problems
This paper introduces a novel approach to learning sparsity-promoting regularizers for solving linear inverse problems. We develop a bilevel optimization framework to select an optimal synthesis operator, denoted as $B$, which regularizes the inverse problem while promoting sparsity in the solution. The method leverages statistical properties of the underlying data and incorporates prior knowledge through the choice of $B$. We establish the well-posedness of the optimization problem, provide theoretical guarantees for the learning process, and present sample complexity bounds. The approach is demonstrated through examples, including compact perturbations of a known operator and the problem of learning the mother wavelet, showcasing its flexibility in incorporating prior knowledge into the regularization framework. This work extends previous efforts in Tikhonov regularization by addressing non-differentiable norms and proposing a data-driven approach for sparse regularization in infinite dimensions.
Code (0)
등록된 구현이 없습니다.
Tasks
Bilevel OptimizationSimilar Papers 제목 키워드 기반
Supervised Learning of Sparsity-Promoting Regularizers for Denoising
We present a method for supervised learning of sparsity-promoting regularizers for image denoising. Sparsity-promoting regularization is a key ingredient in solving modern image reconstruction problems; however, the oper…
Collaborative FilteringDenoisingDictionary LearningImage Denoising+1Learning Sparsity-Promoting Regularizers using Bilevel Optimization
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; how…
Bilevel OptimizationCollaborative FilteringDenoisingDictionary Learning+1Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithms
We propose to learn non-convex regularizers with a prescribed upper bound on their weak-convexity modulus. Such regularizers give rise to variational denoisers that minimize a convex energy. They rely on few parameters (…
Image ReconstructionMRI ReconstructionBilevel learning of l1-regularizers with closed-form gradients(BLORC)
We present a method for supervised learning of sparsity-promoting regularizers, a key ingredient in many modern signal reconstruction problems. The parameters of the regularizer are learned to minimize the mean squared e…
Bilevel OptimizationDenoisingFormLearning a Gaussian Mixture for Sparsity Regularization in Inverse Problems
In inverse problems, it is widely recognized that the incorporation of a sparsity prior yields a regularization effect on the solution. This approach is grounded on the a priori assumption that the unknown can be appropr…
Dictionary Learning