# Efficient algorithms for regularized Poisson Non-negative Matrix Factorization

arXiv: https://arxiv.org/abs/2404.16505 · HTML: https://mlsanity.com/paper/2404.16505/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 60 references that are in the map.
- First posted on arXiv 25 Apr 2024.

## Builds on (most cited first, 14 of 14)

- [How to Escape Saddle Points Efficiently](https://mlsanity.com/paper/1703.00887/) (2017)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Incremental Majorization-Minimization Optimization with Application to Large-Scale Machine Learning](https://mlsanity.com/paper/1402.4419/) (2014)
- [First-order methods almost always avoid saddle points: the case of vanishing step-sizes](https://mlsanity.com/paper/1906.07772/) (2019)
- [The Why and How of Nonnegative Matrix Factorization](https://mlsanity.com/paper/1401.5226/) (2014)
- [Playing with Duality: An Overview of Recent Primal-Dual Approaches for Solving Large-Scale Optimization Problems](https://mlsanity.com/paper/1406.5429/) (2014)
- [A Newton-CG Algorithm with Complexity Guarantees for Smooth Unconstrained Optimization](https://mlsanity.com/paper/1803.02924/) (2018)
- [Accelerated Bregman Proximal Gradient Methods for Relatively Smooth Convex Optimization](https://mlsanity.com/paper/1808.03045/) (2018)
- [Algorithms for Nonnegative Matrix Factorization with the Kullback-Leibler Divergence](https://mlsanity.com/paper/2010.01935/) (2020)
- [Complexity analysis of second-order line-search algorithms for smooth nonconvex optimization](https://mlsanity.com/paper/1706.03131/) (2017)
- [Scalable Recommendation with Poisson Factorization](https://mlsanity.com/paper/1311.1704/) (2013)
- [Primal-Dual Algorithms for Non-negative Matrix Factorization with the Kullback-Leibler Divergence](https://mlsanity.com/paper/1412.1788/) (2014)
- [Provable Bregman-divergence based Methods for Nonconvex and Non-Lipschitz Problems](https://mlsanity.com/paper/1904.09712/) (2019)
- [Fast and Simple Optimization for Poisson Likelihood Models](https://mlsanity.com/paper/1608.01264/) (2016)

## Abstract (arXiv)

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.