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

TL;DR

This paper introduces a structured learning framework for inverse problems that combines classical optimization guarantees with neural network flexibility, demonstrated on EEG imaging.

Contribution

We develop a learned Majorization-Minimization approach that explicitly controls curvature, ensuring stability and convergence in inverse problem solutions.

Findings

Improved accuracy over deep unrolling methods.

Enhanced stability and robustness in EEG imaging.

Better 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.