  We consider the problem of regularized Poisson Non-negative Matrix
Factorization (NMF) problem, encompassing various regularization terms such as
Lipschitz and relatively smooth functions, alongside linear constraints. This
problem holds significant relevance in numerous Machine Learning applications,
particularly within the domain of physical linear unmixing problems. A notable
challenge arises from the main loss term in the Poisson NMF problem being a KL
divergence, which is non-Lipschitz, rendering traditional gradient
descent-based approaches inefficient. In this contribution, we explore the
utilization of Block Successive Upper Minimization (BSUM) to overcome this
challenge. We build approriate majorizing function for Lipschitz and relatively
smooth functions, and show how to introduce linear constraints into the
problem. This results in the development of two novel algorithms for
regularized Poisson NMF. We conduct numerical simulations to showcase the
effectiveness of our approach.
