Zero-Flow Two-Sample Tests

Motivated by the success of modern flow-based generative models in modeling complex data, we study two-sample testing through the lens of flow-based methods. We propose the Zero-Flow Two-Sample Test (ZF2ST), built on the zero-flow criterion, which characterizes distributional equality through a time-reversal antisymmetry of a learnable velocity field. We extend this criterion and further develop the Zero-Flow Discrepancy, an identifying discrepancy that controls the Wasserstein distance, and derive a variational representation in terms of a witness function. This representation naturally leads to a witness-based test whose power is governed by the signal-to-noise ratio (SNR), allowing direct power maximization for witness learning. ZF2ST learns the witness on one data split and performs testing on held-out samples, thereby maintaining Type-I error control and admitting a simple asymptotic null distribution. Experimentally, ZF2ST performs competitively across various synthetic and real-world benchmarks, while showing particularly strong performance in distinguishing image distributions from different sources.