# How fine a change can moments see? A scale law for detecting distribution shift, with a kernel calibration rule

arXiv: https://arxiv.org/abs/2608.01268 · HTML: https://mlsanity.com/paper/2608.01268/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 21 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Failing Loudly: An Empirical Study of Methods for Detecting Dataset Shift](https://mlsanity.com/paper/1810.11953/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Adversary Detection in Neural Networks via Persistent Homology](https://mlsanity.com/paper/1711.10056/) (2017)
- [Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey](https://mlsanity.com/paper/2312.05840/) (2023)

## Abstract (arXiv)

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.