Learning Decision Boundaries for Multidimensional Anomaly Detection

In this article, we attempt to study a problem of identifying outliers from multiple dimensions, terming it as multidimensional anomaly detection. This task assumes that each sample corresponds to heterogeneous discriminative spaces, where each space characterizes distinct semantic information along one dimension. Consequently, the abnormalities exhibited by a sample will vary under these semantically different dimensions. In contrast to traditional anomaly detection, multidimensional anomaly detection offers a more holistic assessment of a sample's abnormalities. However, the heterogeneity of discriminative spaces leads to incomparability of the outputs from different dimensions which is the major difficulty in designing multidimensional anomaly detection methods. This article introduces a novel model, maximum margin multidimensional anomaly detection (ALOE), specifically tailored for multidimensional anomaly detection. ALOE constructs a convex optimization problem with nonlinear constraints. The primary objective is to simultaneously learn multiple decision boundaries, utilizing the maximum margin principle and covariance regularization, while distinguishing between outliers and normal samples under multiple dimensions by capturing the correlation among multiple dimensions. To obtain the optimal decision boundary under each dimension, we devise an alternating optimization method for this convex optimization problem. To validate the effectiveness of ALOE, we conduct extensive experiments on 12 real-world datasets, comparing its performance against 34 anomaly detection methods. The experimental results demonstrate the superior performance of ALOE.