Unrolled Multiplicative Updates for Nonnegative Matrix Factorization applied to Hyperspectral Unmixing

HyperSpectral Unmixing (HSU), the problem of separating mixed spectra of overlapping materials in a hyperspectral image, has motivated dedicated algorithmic developments in the last two decades. On the one hand, traditional model-based algorithms frequently guarantee interpretable results. On the other hand, deep-learning-based approaches are often faster at inference time and may obtain better empirical results. This work utilizes the strengths of both approaches by building on the deep unrolling paradigm. Our contribution is twofold. First, we propose two new algorithms based on deep unrolling of the well-known Multiplicative Updates. The first, coined Non-Adaptive Learned Multiplicative Updates (NALMU), adopts a simple element-wise multiplicative scheme. The second, called Recursive Adaptive Learned Multiplicative Updates (RALMU), has more flexible updates and better take into account the spatial correlations in the abundances. Second, we relate NALMU to the minimization of an explicit cost function under some assumptions. Such guarantees are unique in the HSU field. NALMU and RALMU are tested on astrophysics and remote sensing datasets. They outperform the other deep learning-based HSU algorithms and classical iterative schemes for the endmember estimates and obtain competitive results for the abundance estimates, even when trained in a self-supervised way. The code used in this paper will be made available upon publication.