This work proposes and investigates a novel method for anomaly detection and shows it to be competitive in a variety of Euclidean and non-Euclidean situations. It is based on an extension of the depth quantile functions (DQF) approach. The DQF approach encodes geometric information about a point cloud via functions of a single variable, whereas each observation in a data set is associated with a single such function. Plotting these functions provides a very beneficial visualization aspect. This technique can be applied to any data lying in a Hilbert space. The proposed anomaly detection approach is motivated by the geometric insight of the presence of anomalies in data being tied to the existence of antimodes in the data generating distribution. Coupling this insight with novel theoretical understanding into the shape of the DQFs gives rise to the proposed adaptive DQF (aDQF) methodology. Applications to various data sets illustrate the DQF and aDQF's strong anomaly detection performance, and the benefits of its visualization aspects.

翻译：本文提出并研究了一种新的异常检测方法，并证明其在多种欧几里得与非欧几里得场景中具有竞争力。该方法基于深度分位函数（DQF）方法的扩展。DQF方法通过单变量函数对点云的几何信息进行编码，而数据集中的每个观测值均对应一个这样的函数。绘制这些函数可提供极具价值的可视化视角。该技术适用于任意希尔伯特空间中的数据。所提出的异常检测方法源于几何洞察：数据中异常的存在与数据生成分布中反模态的出现相关。将此洞察与对DQF形状的新理论理解相结合，便诞生了所提出的自适应深度分位函数（aDQF）方法。在多种数据集上的应用展示了DQF与aDQF强大的异常检测性能及其可视化优势。