Convex Regularization Behind Neural Reconstruction
This work is significant for researchers and practitioners in medical imaging and other sensitive applications where interpretability and guaranteed optimality of neural network reconstructions are crucial.
This paper addresses the challenge of non-convexity in neural network-based image reconstruction by introducing a convex duality framework for a two-layer fully-convolutional ReLU denoising network. This framework enables optimal training with convex solvers and provides insights into the network's behavior, showing that weight decay regularization induces path sparsity and prediction involves piecewise linear filtering.
Neural networks have shown tremendous potential for reconstructing high-resolution images in inverse problems. The non-convex and opaque nature of neural networks, however, hinders their utility in sensitive applications such as medical imaging. To cope with this challenge, this paper advocates a convex duality framework that makes a two-layer fully-convolutional ReLU denoising network amenable to convex optimization. The convex dual network not only offers the optimum training with convex solvers, but also facilitates interpreting training and prediction. In particular, it implies training neural networks with weight decay regularization induces path sparsity while the prediction is piecewise linear filtering. A range of experiments with MNIST and fastMRI datasets confirm the efficacy of the dual network optimization problem.