Explainable Learning Based Regularization of Inverse Problems

2025/12/09 by Burger, Martin, Kabri, Samira, Kutyniok, Gitta +2
Computer Science · Engineering · #47A52 #65J20 #65K10 #Adversarial Robustness in Machine Learning #FOS: Mathematics #Numerical Analysis (math.NA) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques

paper · doi:10.48550/arxiv.2512.08758

openalex publication_date 2025/12/09 · openalex created_date 2025/12/11 · openalex updated_date 2026/07/28

Abstract

Machine learning techniques for the solution of inverse problems have become an attractive approach in the last decade, while their theoretical foundations are still in their infancy. In this chapter we want to pursue the study of regularization properties, robustness, convergence rates, and structure of regularizers for inverse problems obtained from different learning paradigms. For this sake we study simple architectures that are explainable in the sense that they allow for a theoretical analysis also in the infinite-dimensional limit. In particular we will advance the study of spectral architectures with new results on convergence rates highlighting the role of the smoothness in the training data set, and a study of adversarial robustness. We can show that adversarial training is actually a convergent regularization method. Moreover, we discuss extensions to frame systems and CNN-type architectures for variational regularizers, where we obtain some results on their structure by carefully designed numerical experiments.

Citations

• DeepInverse: A Python package for solving imaging inverse problems with deep learning
• Flow Matching: Markov Kernels, Stochastic Processes and Transport Plans
• Adversarial flows: A gradient flow characterization of adversarial attacks
• Conditional Wasserstein Distances with Applications in Bayesian OT Flow Matching
• Variational Bayesian Imaging with an Efficient Surrogate Score-based Prior
• Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Medical Image Reconstruction
• Convergent regularization in inverse problems and linear plug-and-play denoisers
• A Variational Perspective on Solving Inverse Problems with Diffusion Models
• Score-Based Diffusion Models as Principled Priors for Inverse Imaging
• Convergent Data-driven Regularizations for CT Reconstruction
• Learning convex regularizers satisfying the variational source condition for inverse problems
• End-to-end reconstruction meets data-driven regularization for inverse problems
• Wasserstein GANs Work Because They Fail (to Approximate the Wasserstein Distance)
• Score-Based Generative Modeling through Stochastic Differential Equations
• TFPnP: Tuning-free Plug-and-Play Proximal Algorithm with Applications to Inverse Imaging Problems
• Regularization of Inverse Problems by Filtered Diagonal Frame Decomposition
• An Algorithm Unrolling Approach to Deep Image Deblurring
• Regularization by Denoising: Clarifications and New Interpretations
• Adversarial Regularizers in Inverse Problems
• Modern Regularization Methods for Inverse Problems
• The Little Engine that Could: Regularization by Denoising (RED)
• Poisson Inverse Problems by the Plug-and-Play scheme
• Explaining and Harnessing Adversarial Examples

Related

• Learned Regularization for Inverse Problems: Insights from a Spectral Model
• Introduction to Regularization and Learning Methods for Inverse Problems
• Let data talk: data-regularized operator learning theory for inverse problems
• Deep Learning for Inverse Problems: Bounds and Regularizers
• Variational Regularization in Inverse Problems and Machine Learning