---
title: "Chazal et al. (2014) prove subsampled persistence-diagram estimators are stable under noise and outliers — not under selection by a structural covariate"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "https://arxiv.org/abs/1406.1901"
source_title: "Subsampling Methods for Persistent Homology"
source_author: "Frédéric Chazal, Brittany Terese Fasy, Fabrizio Lecci, Bertrand Michel, Alessandro Rinaldo, Larry Wasserman"
source_date: "2014-06"
source_quote: "stable with respect to perturbations of the underlying measure"
source_tier: 1
audit_status: "capture-verified — full PDF fetched and read directly via extract_pdf at capture time (arxiv.org/pdf/1406.1901, tls:verified); not independently re-fetched at this promotion pass."
provenance: "Promotion from 10-inbox/raw/2026-07-21-do-persistent-homology-based-neural-network-generalization-diagnostics.md, 2026-07-22 (headless)"
origin: "batch"
derived_from: "10-inbox/raw/2026-07-21-do-persistent-homology-based-neural-network-generalization-diagnostics.md"
date_created: "2026-07-22T00:00:00.000Z"
tags: ["persistent-homology","topological-data-analysis","subsampling","sampling-artifact","statistics","methodology"]
audits: ["2026-07-23 claude-opus-4-8"]
---


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.