先压缩再检验：近线性时间内的强效核检验
Compress Then Test: Powerful Kernel Testing in Near-linear Time
AI总结：
针对现有核双样本检验在时间与功效间的权衡问题，提出基于样本压缩的CTT框架，在近线性时间内保持最优检测边界，实验实现20-200倍加速且不损失功效。
摘要：
核双样本检验为基于$n$个样本点区分任意一对分布提供了强有力的框架。然而，现有的核检验要么以$n^2$的时间运行，要么为了提升运行时间而牺牲过多的检验功效。为解决这些不足，我们引入了“先压缩再检验”（Compress Then Test, CTT），一种基于样本压缩的高功效核检验新框架。CTT通过将每个包含$n$个点的样本压缩成一个小型但可证明高保真的核心集，来廉价地近似一个昂贵的检验。对于标准核和次指数分布，CTT继承了二次时间检验的统计行为——恢复相同的渐近最优检测边界——同时以近线性时间运行。我们将这些进展与更廉价的置换检验相结合，并通过新的功效分析加以论证；改进了低秩近似的时间-质量权衡保证；以及一种快速聚合过程，用于识别特别有判别力的核。在我们使用真实和模拟数据的实验中，CTT及其扩展方法在功效不损失的情况下，比最先进的近似MMD检验提供了20至200倍的加速。
英文摘要：
Kernel two-sample testing provides a powerful framework for distinguishing any pair of distributions based on $n$ sample points. However, existing kernel tests either run in $n^2$ time or sacrifice undue power to improve runtime. To address these shortcomings, we introduce Compress Then Test (CTT), a new framework for high-powered kernel testing based on sample compression. CTT cheaply approximates an expensive test by compressing each $n$ point sample into a small but provably high-fidelity coreset. For standard kernels and subexponential distributions, CTT inherits the statistical behavior of a quadratic-time test -- recovering the same optimal detection boundary -- while running in near-linear time. We couple these advances with cheaper permutation testing, justified by new power analyses; improved time-vs.-quality guarantees for low-rank approximation; and a fast aggregation procedure for identifying especially discriminating kernels. In our experiments with real and simulated data, CTT and its extensions provide 20--200x speed-ups over state-of-the-art approximate MMD tests with no loss of power.
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