Iso-Depth Learning for Anomaly Detection

We consider unsupervised anomaly detection for data that lie near a low-dimensional manifold. Our interest is in detecting points that sit far away from the center of the data cloud but remain close to the manifold. Statistical depth in an ambient space may be inadequate for detecting such points, because Euclidean distance can make points that are far apart along the manifold appear close. Local density scores may also fail to detect such points, because they can have sufficiently many nearby neighbors despite being far from the global center. We propose an Iso-depth framework for anomaly detection that first estimates the intrinsic geometry of the data via ISOMAP and then computes statistical depth in the resulting Euclidean embedding. Iso-depth measures global centrality intrinsically. To facilitate efficient prediction, we train a neural network to approximate the depth function, allowing new observations to be scored with minimal computational cost. We derive an error bound with three components: depth estimation, the ISOMAP embedding, and neural approximation. For halfspace depth, we show how root mean square coordinate error transfers to depth error. We evaluate Iso-depth in controlled synthetic experiments that separate depth estimation, embedding, and out-of-sample prediction errors, and in a real-data application that examines the resulting anomaly rankings.