Discretization of learned NETT regularization for solving inverse problems

TL;DR

This paper extends the analysis of NETT regularization for inverse problems to include discretization effects, demonstrating convergence and providing numerical results in photoacoustic tomography.

Contribution

It introduces a comprehensive framework accounting for discretization in NETT regularization, proving convergence and deriving rates.

Findings

Discretized NETT converges as noise and discretization errors decrease.

Convergence rates are established for the discretized approach.

Numerical results validate theoretical findings in photoacoustic tomography.

Abstract

Deep learning based reconstruction methods deliver outstanding results for solving inverse problems and are therefore becoming increasingly important. A recently invented class of learning-based reconstruction methods is the so-called NETT (for Network Tikhonov Regularization), which contains a trained neural network as regularizer in generalized Tikhonov regularization. The existing analysis of NETT considers fixed operator and fixed regularizer and analyzes the convergence as the noise level in the data approaches zero. In this paper, we extend the frameworks and analysis considerably to reflect various practical aspects and take into account discretization of the data space, the solution space, the forward operator and the neural network defining the regularizer. We show the asymptotic convergence of the discretized NETT approach for decreasing noise levels and discretization errors. Additionally, we derive convergence rates and present numerical results for a limited data problem in photoacoustic tomography.