The statistical distance of conditional distributions is an essential element of generating target data given some data as in video prediction. We establish how the statistical distances between two joint distributions are related to those between two conditional distributions for three popular statistical distances: f-divergence, Wasserstein distance, and integral probability metrics. Such characterization plays a crucial role in deriving a tractable form of the objective function to learn a conditional generator. For Wasserstein distance, we show that the distance between joint distributions is an upper bound of the expected distance between conditional distributions, and derive a tractable representation of the upper bound. Based on this theoretical result, we propose a new conditional generator, the conditional Wasserstein generator. Our proposed algorithm can be viewed as an extension of Wasserstein autoencoders (Tolstikhin et al. 2018) to conditional generation or as a Wasserstein counterpart of stochastic video generation (SVG) model by Denton and Fergus (Denton et al. 2018). We apply our algorithm to video prediction and video interpolation. Our experiments demonstrate that the proposed algorithm performs well on benchmark video datasets and produces sharper videos than state-of-the-art methods.

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.