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.

翻译：我们研究了正则化泊松非负矩阵分解（NMF）问题，该问题涵盖了多种正则化项（如Lipschitz函数和相对光滑函数）以及线性约束。该问题在众多机器学习应用中具有重要意义，特别是在物理线性解混问题领域。一个显著挑战源于泊松NMF问题中的主要损失项是KL散度，该散度非Lipschitz，导致传统基于梯度下降的方法效率低下。在本研究中，我们探索利用块连续上最小化（BSUM）方法克服这一挑战。我们为Lipschitz函数和相对光滑函数构建了适当的优函数，并展示了如何将线性约束引入问题中。由此，我们提出了两种应用于正则化泊松NMF的新算法。通过数值模拟，我们验证了所提方法的有效性。