Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

TL;DR

This paper introduces a novel deep learning approach for inverse problems by leveraging the mathematical connection between proximal operators and Hamilton-Jacobi equations, enabling direct prior learning without inversion.

Contribution

It develops a new deep learning architecture based on the connection between proximal operators and Hamilton-Jacobi PDEs for direct prior learning in inverse problems.

Findings

Efficient high-dimensional inverse problem solutions

Outperforms existing methods in numerical experiments

Direct prior learning without inversion

Abstract

Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they provide a flexible and convenient way to encode priors and build efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., for plug-and-play methods for learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferential of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton-Jacobi partial differential equations (HJ PDEs) to develop novel deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present several numerical results that demonstrate the efficiency of the proposed method in high dimensions.