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

Le Minh Triet Tran, Sarah Reynaud, Ronan Fablet, Adrien Merlini, François Rousseau, Mai Quyen Pham

TL;DR

The paper addresses ill-posed inverse problems by introducing a learned Majorization-Minimization (MM) framework that learns a curvature majorant rather than a full optimizer, preserving MM descent guarantees. The curvature model is parameterized by a lightweight recurrent architecture and can enforce valid quadratic majorants through analytic diagonal bounds for cosine similarity losses and Hessian-vector product-based estimates for general losses. The approach is embedded in a bilevel optimization setup, yielding a provably convergent lower-level solver within the upper-level learning loop. Applied to EEG source imaging, the method achieves superior accuracy, stability, and cross-dataset generalization compared with deep-unrolled and meta-learning baselines, indicating robust optimization geometry transfer across datasets and temporal dynamics.

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.

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

TL;DR

The paper addresses ill-posed inverse problems by introducing a learned Majorization-Minimization (MM) framework that learns a curvature majorant rather than a full optimizer, preserving MM descent guarantees. The curvature model is parameterized by a lightweight recurrent architecture and can enforce valid quadratic majorants through analytic diagonal bounds for cosine similarity losses and Hessian-vector product-based estimates for general losses. The approach is embedded in a bilevel optimization setup, yielding a provably convergent lower-level solver within the upper-level learning loop. Applied to EEG source imaging, the method achieves superior accuracy, stability, and cross-dataset generalization compared with deep-unrolled and meta-learning baselines, indicating robust optimization geometry transfer across datasets and temporal dynamics.

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.
Paper Structure (30 sections, 2 theorems, 44 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 30 sections, 2 theorems, 44 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works
  3. Preliminaries
  4. Problem Setup
  5. Quadratic MM and Majorant Condition
  6. Learning a Convergent MM Solver
  7. MM solver for Cosine Similarity
  8. MM solver for a twice differentiable and gradient Lipschitz loss function
  9. Bilevel Optimization with MM-based Solver
  10. Bilevel optimization
  11. Solver Architecture with Curvature Majorant
  12. Application: ESI problem
  13. Problem: EEG source imaging (ESI)
  14. Data simulation
  15. Head model and forward operator
  16. ...and 15 more sections

Key Result

Proposition 1

For every ${\bf x}(\hbox{\boldmath$\theta$}) \in {\mathbb R}^s$, ${\bf y}\in {\mathbb R}^n$, and ${\bf L} \in {\mathbb R}^{n\times s}$, define the vector $\mathbf{p}({\bf x}(\hbox{\boldmath$\theta$})) \in {\mathbb R}^s$ such that where $\underline{\nu}>0$, $\mu_1 = 5\frac{\|{\bf L}\|^2}{\|{\bf L}{\bf x}\|^2},$ and If there exist $0 <\underline{o} < \overline{o} <+\infty$ such that $\alpha, \rho

Figures (2)

  • Figure 1: Visualizing the estimate result of the methods.
  • Figure 2: Curve of the lower-level cost function on validation data: total loss (loss), loss of data fidelity term (fidelity loss), and loss of regularization term (reg. loss).

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • proof
  • Lemma 1
  • proof