Majorization-Minimization Networks for Inverse Problems: An Application to EEG Imaging

2026-01-23

arXiv.org

AI Summary

This work addresses the instability and convergence difficulties in ill-posed inverse problems such as EEG source imaging by proposing a learning-based majorization-minimization (MM) framework. Operating within a bilevel optimization setting, the method employs a lightweight recurrent neural network to learn structured curvature-majorizing surrogates that satisfy MM conditions, rather than end-to-end learning a full optimizer, thereby guaranteeing strict descent of the objective function at each iteration. By integrating analytical curvature bounds with Hessian-vector product spectral estimation and incorporating cosine similarity constraints to construct diagonal majorizers, the approach ensures theoretical convergence while significantly enhancing robustness and generalization. Experiments demonstrate superior performance over deep unfolding and meta-learning baselines in EEG source imaging, achieving notable improvements in reconstruction accuracy, optimization stability, and cross-dataset generalization.

Abstract

Inverse problems are often ill-posed and require optimization schemes with strong stability and convergence guarantees. While learning-based approaches such as deep unrolling and meta-learning achieve strong empirical performance, they typically lack explicit control over descent and curvature, limiting robustness. We propose a learned Majorization-Minimization (MM) framework for inverse problems within a bilevel optimization setting. Instead of learning a full optimizer, we learn a structured curvature majorant that governs each MM step while preserving classical MM descent guarantees. The majorant is parameterized by a lightweight recurrent neural network and explicitly constrained to satisfy valid MM conditions. For cosine-similarity losses, we derive explicit curvature bounds yielding diagonal majorants. When analytic bounds are unavailable, we rely on efficient Hessian-vector product-based spectral estimation to automatically upper-bound local curvature without forming the Hessian explicitly. Experiments on EEG source imaging demonstrate improved accuracy, stability, and cross-dataset generalization over deep-unrolled and meta-learning baselines.

Problem

• inverse problems
• ill-posed
• optimization
• robustness
• stability

Innovation

• Majorization-Minimization
• inverse problems
• curvature majorant
• bilevel optimization
• EEG source imaging