提出检测分布偏移的尺度定律，指导核方法带宽选择。
How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule
- 基于矩的检测需满足度数下限，由切比雪夫极值问题推导得出。
- 实测表明带宽匹配的核检验在多种场景下AUC超0.95，优于其他方法。
- 适合关注分布检测效率与鲁棒性的研究人员使用。
检测高维嵌入流是否发生分布变化通常依赖于统计量的选择。本文给出一个约束任何基于矩的检测方法的尺度定律：要检测尺度为eps、质量占比为f的特征，需使用次数N* >= log(1/f)/(2 eps)的多项式检验，该结论通过切比雪夫极值问题证明；对于b尺度拓扑结构，构造高斯积分得N* >= 4b-1，因此成本由特征精细度决定而非特征数量。该定律为单边成立：我们构造出一个环形区域，其均值、协方差及所有四阶矩均与实心圆相同，但一阶同调群H_1非零。其实际意义在于提供校准规则：高斯测试函数达到上界，即RBF核对应的RKHS见证，因此该定律可预测MMD检验应采用的带宽——特征尺度。在真实嵌入流上，σ*/eps中位数为1.12（四分位距1.01-1.52，n=26），三种设置三种尺度下，数据驱动带宽使AUC ≥ 0.95。面对针对均值、协方差、k-NN、峰度等统计量优化的对手，仅有带宽匹配的核检验仍能检测到变化。持久性拓扑结果混杂且依赖隐含假设。摘要重要性大于滤波过程：总持续性在FPR 1%时召回率达0.75，而首层持久性景观仅为0.00。真正差异在于成本差距而非检测能力：当持久性有效时，其成本是峰度检验的116倍，而后者至少同样有效。结论并非拓扑摘要无用，而是在此任务中，按定律设定带宽的核检验占优。
原文摘要 · Abstract (English)
Detecting that a stream of high-dimensional embeddings has changed is usually framed as a choice of statistic. We give a scale law that constrains any moment-based choice and test it against topological alternatives. The law: certifying a feature of spatial scale eps carrying mass fraction f requires polynomial tests of degree N* >= log(1/f)/(2 eps), proved via the Chebyshev extremal problem; a Gauss-quadrature construction gives N* >= 4b-1 for a b-scale topology, so cost is set by feature fineness, not feature count. The law is one-sided: we exhibit an annulus whose mean, covariance and all fourth-order moments equal those of a filled disk, yet H_1 is nonzero. Its practical content is a calibration rule. The upper bound is attained by Gaussian test functions, the RKHS witness of an RBF kernel, so the law predicts which bandwidth an MMD test should use: the feature scale. On real embedding streams we measure sigma*/eps with median 1.12 (IQR 1.01-1.52, n=26) over three settings and three scales, and a data-driven bandwidth reaches AUC >= 0.95. Against an adversary optimised against the defender's statistics (mean, covariance, k-NN, kurtosis), only a bandwidth-matched kernel test still detects. For persistent homology the verdict is mixed and depends on choices usually left implicit. The summary matters more than the filtration: total persistence attains recall 0.75 at FPR 1% where the first persistence landscape attains 0.00. What survives is a cost gap, not a power gap: where persistence works it costs 116x kurtosis, which works at least as well. We conclude not that topological summaries are useless, but that on this task a kernel test whose bandwidth the law sets dominates them.
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