比较分布：$\ell_1$几何改进了内核双样本测试
Comparing distributions: $\ell_1$ geometry improves kernel two-sample testing
摘要：
两组观测值是否来自同一分布？这个问题是一个双样本测试。核方法产生了许多吸引人的属性。实际上，最先进的方法使用基于内核的分布代表之间的$L^2$距离来导出它们的检验统计量。在这里，我们表明，$L^p$距离（与$p\geq 1$）这些分布的代表给出的度量空间的分布，表现良好，以检测分布之间的差异，因为他们度量弱收敛。此外，对于解析核，我们表明，$L^1$几何提供了可扩展的计算程序，提高了测试能力。具体来说，我们推导出一个有限维近似的度量给出的$\ell_1$规范的向量，捕捉不同的期望的分析功能评估在空间位置或频率（即功能）。可以选择特征以最大化分布的差异，并给出它们如何不同的可解释指示。使用$\ell_1$范数提供更好的检测，因为代表之间的差异是密集的，因为我们使用分析内核（几乎处处非零）。这些测试是一致的，同时比最先进的基于二次时间内核的测试快得多。基于$\ell_2$规范，对人工和真实世界问题的实验证明了比现有技术更好的功率/时间权衡，并且在某些情况下，甚至比最昂贵的二次时间测试更好的直接功率。
英文摘要：
Are two sets of observations drawn from the same distribution? This problem is a two-sample test. Kernel methods lead to many appealing properties. Indeed state-of-the-art approaches use the $L^2$ distance between kernel-based distribution representatives to derive their test statistics. Here, we show that $L^p$ distances (with $p\geq 1$) between these distribution representatives give metrics on the space of distributions that are well-behaved to detect differences between distributions as they metrize the weak convergence. Moreover, for analytic kernels, we show that the $L^1$ geometry gives improved testing power for scalable computational procedures. Specifically, we derive a finite dimensional approximation of the metric given as the $\ell_1$ norm of a vector which captures differences of expectations of analytic functions evaluated at spatial locations or frequencies (i.e, features). The features can be chosen to maximize the differences of the distributions and give interpretable indications of how they differs. Using an $\ell_1$ norm gives better detection because differences between representatives are dense as we use analytic kernels (non-zero almost everywhere). The tests are consistent, while much faster than state-of-the-art quadratic-time kernel-based tests. Experiments on artificial and real-world problems demonstrate improved power/time tradeoff than the state of the art, based on $\ell_2$ norms, and in some cases, better outright power than even the most expensive quadratic-time tests.
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