A loss framework for calibrated anomaly detection

Given samples from a distribution, anomaly detection is the problem of determining if a given point lies in a low-density region. This paper concerns calibrated anomaly detection, which is the practically relevant extension where we additionally wish to produce a confidence score for a point being anomalous. Building on a classification framework for standard anomaly detection, we show how minimisation of a suitable proper loss produces density estimates only for anomalous instances. These are shown to naturally relate to the pinball loss, which provides implicit quantile control. Finally, leveraging a result from point processes, we show how to efficiently optimise a special case of the objective with kernelised scores. Our framework is shown to incorporate a close relative of the one-class SVM as a special case.