Robust Hypothesis Testing Using Wasserstein Uncertainty Sets

• DOI: 10.48550/arxiv.1805.10611
• Publication Date: 2018-01-01
• Authors: Gao, Rui; Xie, Liyan; Xie, Yao; Xu, Huan
• Keywords: FOS: Computer and information sciences; Computer Science - Machine Learning; Computer Science - Information Theory; Information Theory (cs.IT); Machine Learning (stat.ML); 02 engineering and technology; Machine Learning (cs.LG); Statistics - Machine Learning; Optimization and Control (math.OC); 0202 electrical engineering, electronic engineering, information engineering; FOS: Mathematics; Mathematics - Optimization and Control

We develop a novel computationally efficient and general framework for robust hypothesis testing. The new framework features a new way to construct uncertainty sets under the null and the alternative distributions, which are sets centered around the empirical distribution defined via Wasserstein metric, thus our approach is data-driven and free of distributional assumptions. We develop a convex safe approximation of the minimax formulation and show that such approximation renders a nearly-optimal detector among the family of all possible tests. By exploiting the structure of the least favorable distribution, we also develop a tractable reformulation of such approximation, with complexity independent of the dimension of observation space and can be nearly sample-size-independent in general. Real-data example using human activity data demonstrated the excellent performance of the new robust detector.