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Learned optimization algorithms are promising approaches to inverse problems by leveraging advanced numerical optimization schemes and deep neural network techniques in machine learning. In this paper, we propose a novel deep neural network architecture imitating an extra proximal gradient algorithm to solve a general class of inverse problems with a focus on applications in image reconstruction. The proposed network features learned regularization that incorporates adaptive sparsification mappings, robust shrinkage selections, and nonlocal operators to improve solution quality. Numerical results demonstrate the improved efficiency and accuracy of the proposed network over several state-of-the-art methods on a variety of test problems. 

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In this model, the nonconvex nonsmooth regularization term is parameterized as a structured deep network where the network parameters can be learned from data. We partition these network parameters into two parts: a task-invariant part for the common feature encoder component of the regularization, and a task-specific part to account for the variations in the heterogeneous training and testing data. We train the regularization parameters in a bilevel optimization framework which significantly improves the robustness of the training process and the generalization ability of the network. We conduct a series of numerical experiments using heterogeneous MRI data sets with various undersampling patterns, ratios, and acquisition settings. The experimental results show that our network yields greatly improved reconstruction quality over existing methods and can generalize well to new reconstruction problems whose undersampling patterns/trajectories are not present during training. 

We consider a regression problem, where the correspondence between the input and output data is not available. Such shuffled data are commonly observed in many real world problems. Take flow cytometry as an example: the measuring instruments are unable to preserve the correspondence between the samples and the measurements. Due to the combinatorial nature of the problem, most of the existing methods are only applicable when the sample size is small, and are limited to linear regression models. To overcome such bottlenecks, we propose a new computational framework --- ROBOT --- for the shuffled regression problem, which is applicable to large data and complex models. Specifically, we propose to formulate regression without correspondence as a continuous optimization problem. Then by exploiting the interaction between the regression model and the data correspondence, we propose to develop a hypergradient approach based on differentiable programming techniques. Such a hypergradient approach essentially views the data correspondence as an operator of the regression model, and therefore it allows us to find a better descent direction for the model parameters by differentiating through the data correspondence. ROBOT is quite general, and can be further extended to an inexact correspondence setting, where the input and output data are not necessarily exactly aligned. Thorough numerical experiments show that ROBOT achieves better performance than existing methods in both linear and nonlinear regression tasks, including real-world applications such as flow cytometry and multi-object tracking.