Dimensionality reduction is considered as an important step for ensuring competitive performance in unsupervised learning such as anomaly detection. Non-negative matrix factorization (NMF) is a popular and widely used method to accomplish this goal. But NMF, together with its recent, enhanced version, like graph regularized NMF or symmetric NMF, do not have the provision to include the neighborhood structure information and, as a result, may fail to provide satisfactory performance in presence of nonlinear manifold structure. To address that shortcoming, we propose to consider and incorporate the neighborhood structural similarity information within the NMF framework by modeling the data through a minimum spanning tree. What motivates our choice is the understanding that in the presence of complicated data structure, a minimum spanning tree can approximate the intrinsic distance between two data points better than a simple Euclidean distance does, and consequently, it constitutes a more reasonable basis for differentiating anomalies from the normal class data. We label the resulting method as the neighborhood structure assisted NMF. By comparing the formulation and properties of the neighborhood structure assisted NMF with other versions of NMF including graph regularized NMF and symmetric NMF, it is apparent that the inclusion of the neighborhood structure information using minimum spanning tree makes a key difference. We further devise both offline and online algorithmic versions of the proposed method. Empirical comparisons using twenty benchmark datasets as well as an industrial dataset extracted from a hydropower plant demonstrate the superiority of the neighborhood structure assisted NMF and support our claim of merit.

中文翻译：

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邻域结构辅助非负矩阵分解及其在无监督点异常检测中的应用

降维被认为是确保在无监督学习（例如异常检测）中具有竞争力的重要步骤。非负矩阵分解 (NMF) 是一种流行且广泛使用的方法来实现这一目标。但是 NMF 及其最近的增强版本，如图正则化 NMF 或对称 NMF，没有提供包含邻域结构信息的规定，因此在存在非线性流形结构的情况下可能无法提供令人满意的性能。为了解决这个缺点，我们建议通过最小生成树对数据进行建模来考虑和合并 NMF 框架内的邻域结构相似性信息。促使我们做出选择的是这样一种理解，即在存在复杂数据结构的情况下，最小生成树可以比简单的欧几里德距离更好地近似两个数据点之间的内在距离，因此，它构成了区分异常与正常类数据的更合理的基础。我们将所得方法标记为邻域结构辅助 NMF。通过将邻域结构辅助 NMF 的公式和性质与其他版本的 NMF 进行比较，包括图正则化 NMF 和对称 NMF，很明显，使用最小生成树包含邻域结构信息是一个关键的区别。我们进一步设计了所提出方法的离线和在线算法版本。