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Neural networks have shown tremendous potential for reconstructinghigh-resolution images in inverse problems. The non-convex and opaque nature ofneural networks, however, hinders their utility in sensitive applications suchas medical imaging. To cope with this challenge, this paper advocates a convexduality framework that makes a two-layer fully-convolutional ReLU denoisingnetwork amenable to convex optimization. The convex dual network not onlyoffers the optimum training with convex solvers, but also facilitatesinterpreting training and prediction. In particular, it implies training neuralnetworks with weight decay regularization induces path sparsity while theprediction is piecewise linear filtering. A range of experiments with MNIST andfastMRI datasets confirm the efficacy of the dual network optimization problem.

Convex Regularization Behind Neural Reconstruction