Paper:
Robust Hypothesis Testing Using Wasserstein Uncertainty Sets
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.

Review:
The rebuttal addressed my technical concerns, and also I seemed to have misjudged the size of the contributions at first. My score has been updated.   This paper studies the two-sample non-parametric hypothesis testing problem. Given two collections of probability distribution, the paper studies approximating the best detector against the worst distributions from both collections. A standard surrogate loss approximation is used to upper bound the worst case risk (the maximum of the type I and type II errors) with a convex surrogate function, which is known to yield a good solution.   Some parts of section 3 are novel, but I'm not familiar enough with the literature to pinpoint exactly what. The convex surrogate approximation is a standard relaxation in learning theory. I wish the authors be more clear in this manner.   The main novel contribution seems to be in the setting where the two collections of distributions are wasserstein balls centered around empirical distributions. It that case, Theorem 3 derives a convex relaxation to solving the optimal detector.   The authors then provide experimental justification for their method, showing improved performance on real data from previous algorithms.   The writing was generally good, though I found the high level ideas hard to follow. Please be more clear about which results apply in which setting. Also, the authors really need to justify why considering wasserstein balls around empirical distributions is a good idea. In fact, I have a few technical concerns:  1. What if the two empirical distributions from P1 and P2 have little overlap? It seems  that the hypothesis testing problem becomes trivial  2. Is prop 1 only for the Wasserstein-ball sets? The proof certainly needs this assumption but it's not in the proposition statement.  3. Why in the objective for problem (6) concave? What assumptions on \phi are needed?  4. Please justify the min-max swap in 374   To summarize, given my technical concerns and the fact that the main contribution seems to be deriving a convex relaxation, I'm a bit hesitant to recommend this paper.