Efficient algorithms for regularized Poisson Non-negative Matrix Factorization

TL;DR

This paper introduces two novel algorithms for regularized Poisson Non-negative Matrix Factorization, effectively handling non-Lipschitz KL divergence loss using BSUM, with demonstrated numerical improvements.

Contribution

The paper develops new algorithms for Poisson NMF that incorporate regularization and linear constraints, overcoming non-Lipschitz challenges with BSUM techniques.

Findings

Algorithms outperform existing methods in numerical simulations

Effective handling of non-Lipschitz KL divergence in Poisson NMF

Incorporation of linear constraints enhances model flexibility

Abstract

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.