Mai Quyen Pham

Mai-Quyen Pham, Jérémy Cohen, Thierry Chonavel

Nonnegative Matrix Factorization (NMF) is a fundamental tool in unsupervised learning, widely used for tasks such as dimensionality reduction, feature extraction, representation learning, and topic modeling. Many algorithms have been developed for NMF, including the well-known Multiplicative Updates (MU) algorithm, which belongs to a broader class of majorization-minimization techniques. In this work, we introduce a general second-order optimization framework for NMF under both quadratic and $β$-divergence loss functions. This approach, called Second-Order Majorant (SOM), constructs a local quadratic majorization of the loss function by majorizing its Hessian matrix. It includes MU as a special case, while enabling faster variants. In particular, we propose mSOM, a new algorithm within this class that leverages a tighter local approximation to accelerate convergence. We provide a convergence analysis, showing linear convergence for individual factor updates and global convergence to a stationary point for the alternating version, AmSOM algorithm. Numerical experiments on both synthetic and real data sets demonstrate that mSOM consistently outperforms state-of-the-art algorithms across multiple loss functions.

Le Minh Triet Tran, Sarah Reynaud, Ronan Fablet et al.

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.