Convex Regularization Behind Neural Reconstruction

TL;DR

This paper introduces a convex duality framework for neural networks used in image reconstruction, enabling convex optimization, interpretability, and revealing new insights into training and prediction behaviors.

Contribution

It develops a convex dual network approach for two-layer ReLU denoising networks, enhancing training and interpretability in inverse imaging problems.

Findings

Convex dual network enables optimal training with convex solvers.

Training with weight decay induces path sparsity in the dual network.

Prediction acts as a piecewise linear filter, improving interpretability.

Abstract

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.