The eLinoR project has made significant advances in the theory, algorithms, and models for Generalized Low-Rank Matrix Factorizations (G-LRMFs). The work performed is structured along three core building blocks.
In BB1 (Theory), we established foundational theoretical results. This includes pioneering the first identifiability conditions for the Nonnegative Tucker Decomposition (NTD), crucial for unique and interpretable tensor analysis. We also proved the NP-hardness of computing cone-constrained singular values, delineating computational boundaries, and delivered a definitive tightness analysis of error bounds for the Successive Projection Algorithm (SPA), a cornerstone method in data unmixing, and proved robustness of minimum-volume NMF.
In BB2 (Algorithms), we developed a suite of novel computational tools. We created a state-of-the-art Integer Programming framework for Boolean Matrix Factorization (BMF), providing exact solutions for subproblems. We also introduced the Block Majorization Minimization with Extrapolation (BMMe) framework, yielding the first provably convergent and accelerated algorithms for NMF with β-divergences. Furthermore, we designed practical algorithms for challenging nonlinear models, including decompositions with the ReLU function.
In BB3 (Models), we introduced new G-LRMFs driven by theoretical insights and applications. We developed the first Orthogonal NMF model with the Kullback-Leibler divergence for count data and deep NMF with β-divergences. A key breakthrough was the creation of subtractive mixture models via squaring, a new probabilistic framework exponentially more expressive than traditional mixtures. Application-driven models led to the AMAT algorithm, a novel method for direct exoplanet detection that outperforms existing standards.
The project's outcomes are documented in numerous high-impact publications (e.g. 6 SIAM journal papers), showing progress from core theory to practical algorithms and interdisciplinary applications.
