CONVEX REGULARIZATION BEHIND NEURAL RECONSTRUCTION

Abstract

Neural networks have shown tremendous potential for reconstructing highresolution 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.

1. INTRODUCTION

In the age of AI, image reconstruction has witnessed a paradigm shift that impacts several applications ranging from natural image super-resolution to medical imaging. Compared with the traditional iterative algorithms, AI has delivered significant improvements in speed and image quality, making learned reconstruction based on neural networks widely adopted in clinical scanners and personal devices. The non-convex and opaque nature of deep neural networks however raises serious concerns about the authenticity of the predicted pixels in domains as sensitive as medical imaging. It is thus crucial to understand what the trained neural networks represent, and interpret their reconstruction per pixel for unseen images. Reconstruction is typically cast as an inverse problem, where neural networks are used in different ways to create effective priors; see e.g., (Ongie et al., 2020; Mardani et al., 2018b) and references therein. An important class of methods are denoising networks, which given natural data corrupted by some noisy process Y , aim to regress the ground-truth, noise-free data X * (Gondara, 2016; Vincent et al., 2010) . These networks are generally learned in a supervised fashion, such that a mapping f : Y → X is learned from inputs {y i } n i=1 to outputs {x * i } n i=1 , and then can be used in the inference phase on new samples ŷ to generate the prediction x * = f ( ŷ). The scope of supervised denoising networks is so general that it can cover more structured inverse problems appearing, for example, in compressed sensing. In this case one can easily form a poor (linear) estimate of the ground-truth image that is noisy and then reconstruct via end-to-end denoising networks (Mardani et al., 2018b; Mousavi et al., 2015) . This method has been proven quite effective on tasks such as medical image reconstruction (Mardani et al., 2018b; a; Sandino et al., 2020; Hammernik et al., 2018) , and significantly outperforms sparsity-inducing convex denoising methods, such as total-variation (TV) and wavelet regularization (Candès et al., 2006; Lustig et al., 2008; Donoho, 2006) in terms of both quality and speed. Despite their encouraging results and growing use in clinical settings (Sandino et al., 2020; Hammernik et al., 2018; Mousavi et al., 2015) , little work has explored the interpretation of supervised training of over-parameterized neural networks for inverse problems. Whereas robustness guarantees exist for inverse problems with minimization of convex sparsity-inducing objectives (Oymak & Hassibi, 2016; Chandrasekaran et al., 2012) , there exist no such guarantees for predictions of 1

