Non-negative matrix factorization (NMF) is widely used for dimensionality reduction and interpretable analysis, but standard formulations are unsupervised and cannot directly exploit class labels. Existing supervised or semi-supervised extensions usually incorporate labels only via penalties or graph constraints, still requiring an external classifier. We propose \textit{NMF-LAB} (Non-negative Matrix Factorization for Label Matrix), which redefines classification as the inverse problem of non-negative matrix tri-factorization (tri-NMF). Unlike joint NMF methods, which reconstruct both features and labels, NMF-LAB directly factorizes the label matrix $Y$ as the observation, while covariates $A$ are treated as given explanatory variables. This yields a direct probabilistic mapping from covariates to labels, distinguishing our method from label-matrix factorization approaches that mainly model label correlations or impute missing labels. Our inversion offers two key advantages: (i) class-membership probabilities are obtained directly from the factorization without a separate classifier, and (ii) covariates, including kernel-based similarities, can be seamlessly integrated to generalize predictions to unseen samples. In addition, unlabeled data can be encoded as uniform distributions, supporting semi-supervised learning. Experiments on diverse datasets, from small-scale benchmarks to the large-scale MNIST dataset, demonstrate that NMF-LAB achieves competitive predictive accuracy, robustness to noisy or incomplete labels, and scalability to high-dimensional problems, while preserving interpretability. By unifying regression and classification within the tri-NMF framework, NMF-LAB provides a novel, probabilistic, and scalable approach to modern classification tasks.

翻译：非负矩阵分解（NMF）被广泛用于降维和可解释性分析，但标准形式是无监督的，无法直接利用类别标签。现有的监督或半监督扩展通常仅通过惩罚项或图约束来融入标签，仍然需要一个外部分类器。我们提出 \textit{NMF-LAB}（面向标签矩阵的非负矩阵分解），它将分类问题重新定义为非负矩阵三因子分解（tri-NMF）的逆问题。与同时重构特征和标签的联合NMF方法不同，NMF-LAB 直接将标签矩阵 $Y$ 作为观测值进行分解，而协变量 $A$ 则被视为给定的解释变量。这产生了一个从协变量到标签的直接概率映射，从而将我们的方法与主要建模标签相关性或填补缺失标签的标签矩阵分解方法区分开来。我们的逆问题框架提供了两个关键优势：（i）类别成员概率可直接从分解中获得，无需单独的分类器；（ii）协变量（包括基于核的相似性）可以无缝集成，从而将预测推广到未见样本。此外，未标记数据可以被编码为均匀分布，支持半监督学习。在从小型基准数据集到大规模MNIST数据集的各种数据集上的实验表明，NMF-LAB 实现了有竞争力的预测精度、对噪声或不完整标签的鲁棒性、对高维问题的可扩展性，同时保持了可解释性。通过在三因子NMF框架内统一回归和分类，NMF-LAB 为现代分类任务提供了一种新颖的、概率化的、可扩展的方法。