系统讲解逆问题的正则化与学习方法，连接经典理论与深度学习新范式。
Introduction to Regularization and Learning Methods for Inverse Problems
- 从反演问题出发，建立正则化框架与收敛性理论基础
- 对比传统Tikhonov与稀疏正则化在有限维空间的重建效果
- 引入深度学习中的可学习正则、贝叶斯估计等前沿方法
这些讲义围绕逆问题中的数学概念展开。首先通过微分、去卷积、计算机断层扫描和相位恢复等例子引入逆问题，并讨论适定性及重构方法。第二章介绍希尔伯特空间中的经典正则化理论，包括伪逆与收敛正则化，并探讨如何实现实用的重构算法，重点分析有限维空间下的Tikhonov正则化与稀疏正则化。第三章进入现代深度学习方法，涵盖数据依赖型逆问题求解技术，包括可学习正则化、全可学习贝叶斯估计、后处理策略及插件式方法，聚焦该交叉领域中的一小部分核心进展。
原文摘要 · Abstract (English)
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.
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