Deep Null Space Learning for Inverse Problems: Convergence Analysis and Rates

TL;DR

This paper introduces null space networks with M-regularization for inverse problems, providing theoretical convergence analysis and rates, bridging the gap between empirical success and mathematical justification.

Contribution

It proposes a novel null space network structure with M-regularization, offering the first theoretical convergence and regularization results for two-step deep learning inverse methods.

Findings

Null space networks preserve data consistency.

The approach is proven to be an M-regularization method.

Convergence rates for the method are derived.

Abstract

Recently, deep learning based methods appeared as a new paradigm for solving inverse problems. These methods empirically show excellent performance but lack of theoretical justification; in particular, no results on the regularization properties are available. In particular, this is the case for two-step deep learning approaches, where a classical reconstruction method is applied to the data in a first step and a trained deep neural network is applied to improve results in a second step. In this paper, we close the gap between practice and theory for a new network structure in a two-step approach. For that purpose, we propose so called null space networks and introduce the concept of M-regularization. Combined with a standard regularization method as reconstruction layer, the proposed deep null space learning approach is shown to be a M-regularization method; convergence rates are also derived. The proposed null space network structure naturally preserves data consistency which is considered as key property of neural networks for solving inverse problems.