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.

我们研究了正则化泊松非负矩阵分解（NMF）问题，包括利普希茨和相对平滑函数以及线性约束的各种正则化项。我们利用块递进上界最小化（BSUM）来克服主要损失项为 KL 散度的挑战，构建适当的上界函数，并展示如何引入线性约束进入该问题中。这导致了两种新的正则化泊松 NMF 算法的发展。我们进行了数值模拟展示我们方法的有效性。