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

Eric Weine, Peter Carbonetto, Rafael A. Irizarry, Matthew Stephens

TL;DR

This work introduces log1p Poisson NMF, a non-identity Poisson NMF where the mean is linked to the bilinear factorization via a shifted-log function g(λ; c) = α_c log(1 + λ / c). By varying c, the model smoothly transitions from additive to multiplicative component combinations, enabling more flexible interpretation of count data. The authors provide maximum-likelihood fitting algorithms and a scalable sparse-approximation scheme, along with theoretical results showing bi-concavity and limiting relationships to standard Poisson NMF and Poisson GLM-PCA. Through real and simulated data, they demonstrate that the choice of link function substantially impacts factor structure, sparsity, and interpretability, with practical implications for RNA-seq and text data analyses. The methodology is further supported by reproducible code and a thorough comparison of approximation approaches, positioning log1p NMF as a versatile exploratory tool for count data analysis.

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.

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

TL;DR

This work introduces log1p Poisson NMF, a non-identity Poisson NMF where the mean is linked to the bilinear factorization via a shifted-log function g(λ; c) = α_c log(1 + λ / c). By varying c, the model smoothly transitions from additive to multiplicative component combinations, enabling more flexible interpretation of count data. The authors provide maximum-likelihood fitting algorithms and a scalable sparse-approximation scheme, along with theoretical results showing bi-concavity and limiting relationships to standard Poisson NMF and Poisson GLM-PCA. Through real and simulated data, they demonstrate that the choice of link function substantially impacts factor structure, sparsity, and interpretability, with practical implications for RNA-seq and text data analyses. The methodology is further supported by reproducible code and a thorough comparison of approximation approaches, positioning log1p NMF as a versatile exploratory tool for count data analysis.

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.
Paper Structure (32 sections, 6 theorems, 60 equations, 15 figures, 1 table, 1 algorithm)

This paper contains 32 sections, 6 theorems, 60 equations, 15 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Poisson NMF with the shifted log link function, and connections to existing models
  3. Connections with existing models and methods
  4. Standard Poisson NMF
  5. Poisson GLM-PCA
  6. Frobenius-norm NMF applied to shifted log counts
  7. Algorithms for fitting log1p NMF
  8. Fitting model parameters with block coordinate ascent
  9. Computational complexity and an approximation for sparse data
  10. Accuracy of the sparse computational approximation
  11. Adding row-specific scaling constants
  12. Initialization
  13. Re-scaling inferred loadings and factors
  14. Applications
  15. MCF-7 bulk RNA-seq
  16. ...and 17 more sections

Key Result

Theorem 1

For any fixed $\mathbf{Y} \in \mathbb{N}_{0}^{n \times p}$, for all $\mathbf{L} \in \mathbb{R}_{\geq 0}^{n \times K}$ and $\mathbf{F} \in \mathbb{R}_{\geq 0}^{p \times K}$,

Figures (15)

  • Figure 1: Plots of the link function $g(\lambda; c) = \alpha_c \times \log(1 + \lambda / c)$ for various values of $c$.
  • Figure 2: Scaling of computational complexity of equations \ref{['eq:log1p_ll']} and \ref{['eq:approx_log1p_ll']} with data characteristics and choice of $K$. The y-axes are the ratio of the computational complexity of the specified calculation relative to the complexity of computing the log-likelihood of the standard Poisson NMF model, where $1$ indicates the two calculations have the same complexity. (A) Scaling with respect to $n$ with all other variables fixed (note that this is equivalent to scaling with $m$). (B) Scaling with $K$ with all other variables fixed. (C) Scaling with sparsity $\left(\left[1 - \frac{\omega}{nm}\right] \cdot 100\%\right)$ with all other variables fixed.
  • Figure 3: Likelihood ratio (likelihood of data when optimized with the approximate objective divided by likelihood of data when optimized with the exact objective) of factor models fit with $K = 5$ and varying settings of $c$. All data are generated with $n = m = 500$ and $K = 5$. (A) Likelihood ratios when data are generated from the log1p model with $c = 10^{-3}$. (B) Likelihood ratios when data are generated from the log1p model with $c = 1$. (C) Likelihood ratios when data are generated from the log1p model with $c = \infty$.
  • Figure 4: Combined figure and table for the MCF-7 analysis. (A-B) Visual representation of fitted $\mathbf{L}$ matrices for the topic model and log1p NMF with $c = 1$. Each column represents a row of $\mathbf{L}$, where each color corresponds to a column of $\mathbf{L}$. (C-D) Scatterplots of factors $2$ and $3$. Each point corresponds to a single gene (row of $\mathbf{F}$). Points are colored based on results of differential expression using DESeq2 love2014moderated. Points in green have Benjamini-Hochberg benjamini1995controlling adjusted p-values $< 0.01$ and log2FC $> 1$ in both the RA and TGF-$\beta$ groups. Points in orange meet these conditions in only the RA group, points in blue meet these conditions only in the TGF-$\beta$ group, and points in grey meet these conditions in neither group. (Table): Top $10$ genes in each column of $\mathbf{F}$ from the fitted models.
  • Figure 5: Combined figure and table for the cell type associated factors of the Pancreas analysis. (A-B) Visual representation of fitted $\mathbf{L}$ matrices for the log1p NMF with $c = 1$ and $c = \infty$, grouped by cell type. Each column represents a row of $\mathbf{L}$, where each color corresponds to a column of $\mathbf{L}$. (Table) Top genes for the factors of each model, excluding treatment associated factors.
  • ...and 10 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Lemma 6
  • Lemma 7