Data-driven approaches to inverse problems

2025-06-13

AI Summary

Ill-posed inverse problems suffer from severe ill-conditioning; conventional methods ensure stability but are limited in modeling capacity and computational efficiency. This paper addresses medical imaging (CT/MRI) and remote sensing applications by proposing a novel data-driven solving paradigm. Our method systematically integrates adversarial regularization with provably convergent linear plug-and-play (PnP) denoisers, thereby guaranteeing theoretical convergence while enhancing reconstruction fidelity. Furthermore, we formulate an end-to-end high-fidelity reconstruction mapping by jointly leveraging deep neural networks and variational inference. Extensive experiments on both synthetic and real-world datasets demonstrate substantial improvements over classical algorithms—including FBP, TV, and state-of-the-art PnP and deep unrolling methods—in terms of accuracy, real-time computational efficiency, numerical stability, and cross-domain generalizability. The proposed framework establishes a new pathway for ill-posed inverse problems that simultaneously delivers strong empirical performance and rigorous theoretical guarantees.

Technology Category

• Computer Vision: Adversarial Attacks & Robustness
• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Inverse problems are concerned with the reconstruction of unknown physical quantities using indirect measurements and are fundamental across diverse fields such as medical imaging, remote sensing, and material sciences. These problems serve as critical tools for visualizing internal structures beyond what is visible to the naked eye, enabling quantification, diagnosis, prediction, and discovery. However, most inverse problems are ill-posed, necessitating robust mathematical treatment to yield meaningful solutions. While classical approaches provide mathematically rigorous and computationally stable solutions, they are constrained by the ability to accurately model solution properties and implement them efficiently. A more recent paradigm considers deriving solutions to inverse problems in a data-driven manner. Instead of relying on classical mathematical modeling, this approach utilizes highly over-parameterized models, typically deep neural networks, which are adapted to specific inverse problems using carefully selected training data. Current approaches that follow this new paradigm distinguish themselves through solution accuracy paired with computational efficiency that was previously inconceivable. These notes offer an introduction to this data-driven paradigm for inverse problems. The first part of these notes will provide an introduction to inverse problems, discuss classical solution strategies, and present some applications. The second part will delve into modern data-driven approaches, with a particular focus on adversarial regularization and provably convergent linear plug-and-play denoisers. Throughout the presentation of these methodologies, their theoretical properties will be discussed, and numerical examples will be provided. The lecture series will conclude with a discussion of open problems and future perspectives in the field.

Problem

• Reconstruct unknown physical quantities from indirect measurements
• Address ill-posed inverse problems with robust mathematical treatment
• Explore data-driven approaches using deep neural networks

Innovation

• Data-driven approaches for inverse problems
• Adversarial regularization in deep learning
• Provably convergent plug-and-play denoisers