Deep synthesis regularization of inverse problems

TL;DR

This paper introduces deep synthesis regularization (DESYRE), a novel method that combines deep learning flexibility with rigorous theoretical guarantees for inverse problems, supported by convergence analysis and numerical validation.

Contribution

It proposes DESYRE, integrating neural networks as synthesis operators with proven convergence and rates, bridging deep learning and classical regularization theories.

Findings

Convergence and convergence rates are established for DESYRE.

Numerical experiments demonstrate the effectiveness of the proposed method.

A strategy for constructing synthesis networks within analysis-synthesis frameworks is provided.

Abstract

Recently, a large number of efficient deep learning methods for solving inverse problems have been developed and show outstanding numerical performance. For these deep learning methods, however, a solid theoretical foundation in the form of reconstruction guarantees is missing. In contrast, for classical reconstruction methods, such as convex variational and frame-based regularization, theoretical convergence and convergence rate results are well established. In this paper, we introduce deep synthesis regularization (DESYRE) using neural networks as nonlinear synthesis operator bridging the gap between these two worlds. The proposed method allows to exploit the deep learning benefits of being well adjustable to available training data and on the other hand comes with a solid mathematical foundation. We present a complete convergence analysis with convergence rates for the proposed deep synthesis regularization. We present a strategy for constructing a synthesis network as part of an analysis-synthesis sequence together with an appropriate training strategy. Numerical results show the plausibility of our approach.