用可学习的曲率上界提升脑电成像逆问题求解稳定性
Majorization-Minimization Networks for Inverse Problems: An Application to EEG Imaging
- 通过轻量递归网络学习曲率上界，保持经典优化保证
- 在脑电成像任务中精度与跨数据集泛化性优于深度展开方法
- 无需显式计算海森矩阵即可自动估计局部曲率上限
逆问题通常病态，需具备强稳定性和收敛性保障的优化方法。尽管基于学习的方法如深度展开和元学习表现优异，但通常缺乏对下降步长和曲率的显式控制，限制了鲁棒性。本文提出一种基于双层优化框架的可学习主要化-最小化（MM）方法，不直接学习完整优化器，而是学习一个结构化的曲率上界函数来约束每一步的MM迭代，同时保留经典MM的下降性质。该上界由轻量级循环神经网络参数化，并显式满足有效的MM条件。对于余弦相似度损失，我们推导出显式曲率边界，得到对角上界；当解析边界不可得时，采用高效的海森向量积谱估计法，在不显式构造海森矩阵的前提下自动上界局部曲率。在脑电源成像实验中，该方法显著提升了准确性、稳定性和跨数据集泛化能力，优于深度展开和元学习基线。
原文摘要 · Abstract (English)
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.
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