Introduction to Regularization and Learning Methods for Inverse Problems

TL;DR

This paper provides an overview of inverse problems, covering classical regularization techniques and modern deep-learning approaches, highlighting their mathematical foundations and practical algorithms.

Contribution

It offers a comprehensive introduction to both traditional and contemporary methods for solving inverse problems, emphasizing the intersection with machine learning.

Findings

Classical regularization methods like Tikhonov are effective in finite-dimensional spaces.

Deep learning approaches enable data-dependent solutions for inverse problems.

The integration of machine learning techniques enhances reconstruction quality and efficiency.

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.