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This paper introduces a novel two-sample testing framework leveraging generative transport by constructing a stochastic interpolant between two empirical distributions. The core idea is that the law of this interpolant is time-reflection invariant if the distributions coincide, enabling a test based on the Jensen-Shannon divergence of marginals at symmetric time points. The proposed method achieves minimax separation rates and empirically matches or surpasses state-of-the-art kernel tests, even for densities with no lower bounds.
Generative transport reveals a hidden time-reflection symmetry in interpolants, enabling exact, non-parametric two-sample testing without density estimation.
Deciding whether two samples come from the same distribution is a classical problem in statistics, and generative transport offers a new way to approach it. We build a stochastic interpolant directly between the two samples and observe that, for a symmetric schedule, its law is invariant under the time reflection $t \mapsto 1-t$ whenever the two distributions coincide. We therefore test whether the marginals at times t and 1-t agree by computing their Jensen--Shannon divergence. Both marginals are explicit mixtures over all cross-pairs of observations, so nothing is learned, and permutation calibration gives an exact finite-sample level. For Gaussian noise, this divergence equals a time integral that pairs the reflection defects of the velocity field and of the score, so the test compares transport dynamics rather than endpoints alone. With a narrow-plus-broad noise design, the test attains the minimax separation rate n^{-2s/(4s+d)} over bounded, compactly supported densities whose difference has Sobolev smoothness s>3d/4, with no lower bound on the densities. Fusing a dyadic grid of noise scales through their permutation ranks, without sample splitting, preserves exact level and adapts to unknown s at an iterated-logarithmic cost. Empirically, the test matches or outperforms state-of-the-art kernel two-sample tests.
