Weighted subspace anomaly detection in high-dimensional space

Abstract

Anomaly detection aims at finding anomalies deviating from the normal data patterns. Virtually all anomaly detection methods create a model of the normal patterns before finding anomalies. In high-dimensional scenarios, due to the curse of dimensionality, it is difficult to construct the model of normal patterns in the full dimensional space. Subspace methods assuming that data can be characterized by low-dimensional manifolds have attracted a great deal of research. However, in unsupervised setting, unlabeled data is composed of both the normal and the abnormal data. The existence of anomalies might affect the establishment of the underlying normal subspaces. The undetermined number of the underlying subspaces also brings difficulties in subspace selection. To tackle the aforementioned problems, we come up with a weighted subspace anomaly detection (WSAD) method. We utilize correntropy to construct an objective function to mitigate the influence of the anomalies, which can be regarded as a weighting method for different data. Besides, we introduce an auxiliary variable with block sparsity regularization to achieve adaptive subspace selection, which can be regarded as a weighting method for different subspaces. After the normal underlying subspaces being established, we define the outlier scores by considering the deviation from the underlying subspaces, the local outlier score within subspaces, and the subspace scale. We use the half-quadratic theory to transform the optimization problem defined in WSAD, and apply alternating optimization to solve the transformed problem. Theoretically, we prove the convergence of the optimization algorithm. Experimentally, we demonstrate the effectiveness of the proposed method on both synthetic data and real datasets.