基于 Tweedie 和负二项成本函数的传统和凸 NMF 的 MM 算法及实证评估
MM-algorithms for traditional and convex NMF with Tweedie and Negative Binomial cost functions and empirical evaluation
AI总结：
本文提出了一种统一框架，用于传统和凸 NMF 在 Tweedie 和负二项分布假设下，并通过实证评估展示了其在高类别数场景下的优势。
摘要：
非负矩阵分解（NMF）是一种广泛用于无监督学习和特征提取的工具，应用范围从基因组学到文本分析和信号处理。标准的 NMF 模型通常是在高斯或泊松噪声假设下推导的，这可能不足以处理表现出过分散或复杂均值-方差关系的数据。在本文中，我们开发了一个统一的框架，用于在广泛分布假设下传统和凸 NMF，包括负二项和 Tweedie 模型，并强调了 Tweedie 与 β-分歧之间的联系。使用主要化-最小化方法，我们推导出所有考虑模型的乘法更新规则，并为具有泊松和负二项成本函数的凸 NMF 提供新的更新方法。我们为所有考虑的模型提供了统一的实现，包括几种凸 NMF 模型的首次实现。在突变和词频数据上的实证评估表明，噪声模型的选择对模型拟合和特征恢复有重大影响，并且在类别数较大的情况下，凸 NMF 可以提供一种高效且稳健的替代传统 NMF 的方法。我们的提出的更新的代码可在 R 包 nmfgenr 中找到，并可在 https://github.com/MartaPelizzola/nmfgenr 上获取。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
