通过生成过程进行双样本检验
Two-Sample Testing via Generative Processes
AI总结：
该研究提出一种基于生成式传输的双样本检验方法，通过构建随机插值器并利用时间反射对称性，计算边际分布的Jensen-Shannon散度，实现了精确有限样本水平与最优分离速率，并在实验中优于现有核方法。
摘要：
判断两个样本是否来自同一分布是统计学中的一个经典问题，而生成式传输（generative transport）提供了一种新的处理方式。我们直接在两个样本之间构建一个随机插值器（stochastic interpolant），并观察到，对于对称的时间调度，当两个分布相同时，其分布在时间反射 $t \mapsto 1-t$ 下保持不变。因此，我们通过计算时间 $t$ 和 $1-t$ 处的边际分布之间的 Jensen-Shannon 散度来检验它们是否一致。这两个边际分布都是所有观测值交叉对的显式混合，因此无需学习任何内容，而排列校准（permutation calibration）提供了精确的有限样本水平。对于高斯噪声，该散度等于一个时间积分，该积分将速度场和分数的反射缺陷配对，因此该检验比较的是传输动力学，而非仅比较端点。通过窄加宽（narrow-plus-broad）的噪声设计，该检验在密度有界且紧支撑、其差异具有 Sobolev 光滑度 $s > 3d/4$ 的情况下，达到了极小极大分离速率 $n^{-2s/(4s+d)}$，且对密度没有下界要求。通过排列秩（permutation ranks）融合噪声尺度的二进网格（dyadic grid），无需样本分割，即可保持精确水平，并以迭代对数成本适应未知的 $s$。实验上，该检验匹配或优于最先进的核双样本检验。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
