Non-negative matrix factorisation (NMF) is a widely used tool for unsupervised learning and feature extraction, with applications ranging from genomics to text analysis and signal processing. Standard formulations of NMF are typically derived under Gaussian or Poisson noise assumptions, which may be inadequate for data exhibiting overdispersion or other complex mean-variance relationships. In this paper, we develop a unified framework for both traditional and convex NMF under a broad class of distributional assumptions, including Negative Binomial and Tweedie models, where the connection between the Tweedie and the $β$-divergence is also highlighted. Using a Majorize-Minimisation approach, we derive multiplicative update rules for all considered models, and novel updates for convex NMF with Poisson and Negative Binomial cost functions. We provide a unified implementation of all considered models, including the first implementations of several convex NMF models. Empirical evaluations on mutational and word count data demonstrate that the choice of noise model critically affects model fit and feature recovery, and that convex NMF can provide an efficient and robust alternative to traditional NMF in scenarios where the number of classes is large. The code for our proposed updates is available in the R package nmfgenr and can be found at https://github.com/MartaPelizzola/nmfgenr.

翻译：非负矩阵分解（NMF）是无监督学习和特征提取的常用工具，其应用范围涵盖基因组学、文本分析和信号处理等领域。传统的NMF模型通常基于高斯或泊松噪声假设推导，这些假设可能不适用于呈现过离散或其他复杂均值-方差关系的数据。本文针对包括负二项分布和Tweedie模型在内的广泛分布假设，构建了传统NMF与凸NMF的统一框架，并特别强调了Tweedie模型与$β$-散度之间的关联。通过采用Majorize-Minimisation优化方法，我们推导了所有考虑模型的乘法更新规则，并针对泊松与负二项损失函数的凸NMF提出了创新性更新算法。我们提供了所有模型的统一实现方案，其中包含多个凸NMF模型的首次实现。在突变数据和词频数据上的实证评估表明：噪声模型的选择显著影响模型拟合效果与特征恢复能力；在类别数量较多的场景中，凸NMF能够为传统NMF提供高效且稳健的替代方案。所提更新算法的代码已集成于R软件包nmfgenr中，可通过https://github.com/MartaPelizzola/nmfgenr获取。