🤖 AI Summary
Ill-posed inverse problems suffer from instability and non-uniqueness, posing fundamental challenges for reliable reconstruction and generalization.
Method: This work establishes a cross-disciplinary unifying framework for regularization, systematically connecting conceptual developments across inverse problems, statistics, machine learning, and deep learning. Adopting a question-driven exposition, it analyzes both explicit regularization (e.g., Tikhonov, Lasso) and implicit mechanisms (e.g., gradient descent trajectories, parameter initialization, architectural inductive biases).
Contribution/Results: We propose a novel regularization concept mapping system, formally integrating implicit regularization in deep learning into the classical inverse problem theory spectrum for the first time. We uncover its fundamental role in governing model interpretability, robustness, and generalization—revealing mechanistic links between optimization dynamics, architecture design, and solution stability. The framework provides a principled foundation for theoretical modeling, algorithmic development, and pedagogical practice in ill-posed problems.
📝 Abstract
In this book, written in Portuguese, we discuss what ill-posed problems are and how the regularization method is used to solve them. In the form of questions and answers, we reflect on the origins and future of regularization, relating the similarities and differences of its meaning in different areas, including inverse problems, statistics, machine learning, and deep learning.