Big in Japan: Regularizing Networks for Solving Inverse Problems.

Area of Science:

• Mathematics
• Computer Science
• Image Processing

Background:

• Deep neural networks show promise for inverse problems and image reconstruction.
• Mathematical analysis for neural network-based inverse problem solutions is largely underdeveloped.

Purpose of the Study:

• To introduce and rigorously analyze deep regularizing neural networks (RegNets).
• To establish convergence and derive convergence rates for RegNets in solving inverse problems.

Main Methods:

• Proposed deep regularizing neural networks (RegNets) of the form B_α + N_{θ(α)}B_α.
• Analyzed convergence properties and derived quantitative error estimates.
• Validated results against existing methods and numerical experiments.

Main Results:

• Demonstrated that RegNets constitute a convergent regularization method for inverse problems.
• Derived convergence rates contingent on distance function decay.
• Showcased RegNets outperforming classical regularization and null space networks in sparse data tomography.

Conclusions:

• RegNets offer a mathematically sound framework for inverse problem solving using deep learning.
• The proposed method provides improved performance and convergence guarantees.
• RegNets represent a significant advancement in neural network applications for image reconstruction.