Abstract: Anomaly detection is the task of finding abnormal data that are distinct from normal behavior. Current deep learning-based anomaly detection methods train neural networks with normal data alone and calculate anomaly scores based on the trained model. In this work, we formalize current practices, build a theoretical framework of anomaly detection algorithms equipped with an objective function and a hypothesis space, and establish a desirable property of the anomaly detection algorithm, namely, admissibility. Admissibility implies that optimal autoencoders for normal data yield a larger reconstruction error for anomalous data than that for normal data on average. We then propose a class of admissible anomaly detection algorithms equipped with an integral probability metric-based objective function and a class of autoencoders, Lipschitz continuous autoencoders. The proposed algorithm for Wasserstein distance is implemented by minimizing an approximated Wasserstein distance with a penalty to enforce Lipschitz continuity with respect to Wasserstein distance. Through ablation studies, we demonstrate the efficacy of enforcing Lipschitz continuity of the proposed method. The proposed method is shown to be more effective in detecting anomalies than existing methods via applications to network traffic and image datasets.