提出新方法检测两组数据分布是否显著接近，适用于图像等复杂数据。
Are Two Datasets Close Enough With Statistical Significance? A Kernel Distributional Closeness Testing Approach
- 用自适应范数修正MMD，提升对分布差异的敏感度
- 理论证明新方法检验功效更高且误报率可控
- 适合需要严格比较数据分布的研究者使用
两个分布是否在统计意义上相近？分布接近性检验（DCT）通过检验分布间距离是否至少为ε来形式化这一问题。现有方法主要针对离散空间定义的分布，如使用总变差，限制了其在图像等复杂数据上的应用。为扩展至更广泛数据类型，自然思路是引入最大均值差异（MMD），一种衡量复杂分布差异的强大指标。然而实验表明，多个分布对可能具有相同的MMD值，却在再生核希尔伯特空间（RKHS）中具有不同范数，导致有限样本下可区分性不同，反映实际接近程度不一致，使MMD在DCT中信息量不足。为此，我们设计了一种新分布差异度量——范数自适应MMD（NAMMD），通过使用分布的RKHS范数对MMD值进行缩放。基于NAMMD的渐近分布，我们提出了基于NAMMD的DCT方法，以评估分布对的接近水平。理论上证明，该方法相比基于MMD的DCT具有更高的检验功效，同时保持有界的第一类错误率。大量实验在合成噪声和真实图像等多种数据上验证了该结论。代码已公开于https://github.com/zhijianzhouml/NAMMD。
原文摘要 · Abstract (English)
Are two distributions close to each other with statistical significance? Distribution closeness testing (DCT) formalizes this question by testing whether the distance between a distribution pair is at least epsilon-far. Existing DCT methods mainly measure discrepancies between distribution pairs defined on discrete spaces, for example using total variation, which limits their application to complex data such as images. To extend DCT to more types of data, a natural idea is to introduce maximum mean discrepancy (MMD), a powerful measure of distributional discrepancy between complex distributions, into DCT scenarios. However, empirical results indicate that many distribution pairs can have the same MMD value despite having different norms in the same reproducing kernel Hilbert space (RKHS). These pairs may exhibit different finite-sample distinguishability and reflect different practical closeness levels, making MMD less informative for DCT. To mitigate this issue, we design a new measure of distributional discrepancy, norm-adaptive MMD (NAMMD), which scales the MMD value using the RKHS norms of distributions. Based on the asymptotic distribution of NAMMD, we propose NAMMD-based DCT to assess the closeness level of a distribution pair. Theoretically, we prove that NAMMD-based DCT has higher test power than MMD-based DCT while maintaining bounded type-I error. This is further validated by extensive experiments on multiple types of data, including synthetic noise and real images. Our code is available at https://github.com/zhijianzhouml/NAMMD.
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