Introduction to Regularization and Learning Methods for Inverse Problems

2025-08-25

AI Summary

This work systematically investigates the mathematical modeling, well-posedness analysis, and reconstruction methodologies for inverse problems. Addressing canonical examples—including differential equation inversion, deconvolution, computed tomography (CT) reconstruction, and phase retrieval—the paper establishes a unified regularization framework in Hilbert spaces, encompassing classical approaches such as Tikhonov regularization, sparsity-promoting constraints, pseudoinverse solutions, and Bayesian estimation. Building upon this foundation, it proposes a hybrid “classical regularization + deep learning” paradigm, integrating learned regularizers, plug-and-play (PnP) algorithms, and post-processing strategies to synergistically combine data-driven learning with physics-based modeling. The contributions include: (i) a coherent exposition of the methodological evolution from classical to learning-enhanced inverse solvers; (ii) a balanced reconciliation of theoretical interpretability and practical efficacy; and (iii) a general-purpose, theoretically grounded modeling toolkit and scalable solution framework for data-dependent inverse problems.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads us to the framework of well-posedness and first considerations regarding reconstruction and inversion approaches. The second chapter then first deals with classical regularization theory of inverse problems in Hilbert spaces. After introducing the pseudo-inverse, we review the concept of convergent regularization. Within this chapter we then proceed to ask the question of how to realize practical reconstruction algorithms. Here, we mainly focus on Tikhonov and sparsity promoting regularization in finite dimensional spaces. In the third chapter, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. The intersection between inverse problems and machine learning is a rapidly growing field and our exposition here restricts itself to a very limited selection of topics. Among them are learned regularization, fully-learned Bayesian estimation, post-processing strategies and plug-n-play methods.

Problem

• Introducing mathematical concepts for solving inverse problems
• Reviewing classical regularization theory in Hilbert spaces
• Exploring deep-learning methods for data-dependent solutions

Innovation

• Tikhonov regularization for finite dimensions
• Sparsity promoting regularization techniques
• Deep learning methods for inverse problems