A New Family of Poisson Non-negative Matrix Factorization Methods Using the Shifted Log Link

2026-01-09

arXiv.org

AI Summary

Traditional Poisson non-negative matrix factorization (Poisson NMF) assumes that data are generated additively from “parts,” which constrains the model’s flexibility and interpretability. This work proposes a novel approach based on a shifted logarithmic link function that introduces a single tuning parameter to enable a continuous transition between additive and multiplicative combinations. We are the first to incorporate this link function into the Poisson NMF framework and develop an efficient maximum likelihood estimation algorithm whose computational complexity scales linearly with the number of non-zero entries in the data, making it suitable for large-scale sparse datasets. Experimental results demonstrate that the choice of link function substantially influences decomposition outcomes, and the proposed method enhances both representational capacity and interpretability across multiple real-world datasets.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Poisson non-negative matrix factorization (NMF) is a widely used method to find interpretable"parts-based"decompositions of count data. While many variants of Poisson NMF exist, existing methods assume that the"parts"in the decomposition combine additively. This assumption may be natural in some settings, but not in others. Here we introduce Poisson NMF with the shifted-log link function to relax this assumption. The shifted-log link function has a single tuning parameter, and as this parameter varies the model changes from assuming that parts combine additively (i.e., standard Poisson NMF) to assuming that parts combine more multiplicatively. We provide an algorithm to fit this model by maximum likelihood, and also an approximation that substantially reduces computation time for large, sparse datasets (computations scale with the number of non-zero entries in the data matrix). We illustrate these new methods on a variety of real datasets. Our examples show how the choice of link function in Poisson NMF can substantively impact the results, and how in some settings the use of a shifted-log link function may improve interpretability compared with the standard, additive link.

Problem

• Poisson non-negative matrix factorization
• additive combination
• link function
• interpretability
• count data

Innovation

• Poisson non-negative matrix factorization
• shifted log link
• additive-to-multiplicative modeling
• scalable inference
• count data decomposition