Chazal et al. (2014) prove subsampled persistence-diagram estimators are stable under noise and outliers — not under selection by a structural covariate

Chazal, Fasy, Lecci, Michel, Rinaldo & Wasserman, "Subsampling Methods for Persistent Homology" (arXiv:1406.1901; ICML 2015), anchor the statistical theory behind subsampling persistence diagrams and landscapes. Their estimators, built from i.i.d. random subsamples of a point cloud, are proven "stable with respect to perturbations of the underlying measure" and "robust to the presence of outliers." The stability results bound instability that arises from distributional noise — how much a subsample's estimate can drift because the underlying data is noisy or the subsample happens to miss some structure by chance.

Nothing in this framework addresses instability from which systematic subset of a data structure gets sampled — e.g., sampling only the top-degree nodes of a network, as opposed to drawing i.i.d. at random. That is a different threat model: selection bias by a structural covariate, the specific failure mode claim-hub-selection-artifact-can-reverse-network-breakpoint-signal documents reversing an inferred trend's sign. Chazal et al.'s theorems do not rule this out; the question simply was not theirs to answer.

This matches the vault's own prior framing of the gap in question-tda-neural-net-sampling-artifact-risk — persistence-diagram stability theorems bound instability under noise, not necessarily under selection bias — and grounds one leg of observation-hub-selection-artifact-absent-by-design-in-founding-ph-generalization-papers. Compare claim-stolz-2023-landmark-selection-rules-trade-density-bias-for-noise-sensitivity, which surveys the specific landmark-selection rules this stability theory sits above.