Do persistent-homology-based neural-network generalization diagnostics share the Byzantine trade-network study's hub-selection / sampling-artifact vulnerability?

This capture directly researches the open question logged at question-tda-neural-net-sampling-artifact-risk: whether the two founding neural-network persistent-homology (PH) generalization diagnostics — claim-birdal-2021-persistent-homology-dimension-bounds-generalization and claim-gutierrez-fandino-2021-persistence-diagram-distance-tracks-generalization — inherit the specific failure mode the Byzantine trade-network study caught in itself: a hub-selection artifact, where sampling a network by a structural covariate (node degree) rather than at random can reverse the sign of an inferred signal (claim-hub-selection-artifact-can-reverse-network-breakpoint-signal). Method was to read both NN-PH papers' own method/algorithm sections directly (full PDF, not abstract only), plus two Tier-1 papers from the general TDA subsampling-stability literature.

Claim: Birdal et al.'s persistent-homology-dimension estimator subsamples training iterates uniformly at random, and documents a known small-sample estimation bias rather than a sign-reversing artifact

The PHD estimator in Birdal, Lou, Guibas & Şimşekli (2021) is computed by Algorithm 1, which repeatedly draws a subset of size n from the full set of recorded training iterates and fits a power law across increasing n; the algorithm's own comment labels this step explicitly: "Wn ← sample(W, n) // random sampling." The paper's experimental section confirms the same procedure in prose: "For varying n, we randomly pick n of last iterates and compute Eα0." This is uniform random sampling over the trajectory, not selection by any structural property (e.g., weight magnitude, layer, or training-loss rank) analogous to the Byzantine paper's degree-based hub selection.

The paper does report a known estimator bias tied to sample size: in its synthetic ground-truth ablation (Section 5.2), "as n increases our estimator becomes smoother and well approximates the GT up to a slight over-estimation, a repeatedly observed phenomenon" (citing the intrinsic-dimension-estimation survey Campadelli et al. 2015). This is a magnitude bias that shrinks monotonically as sample size grows, not a sign-reversing artifact tied to which points are chosen — a materially different failure mode from the Byzantine study's hub-selection problem, where the direction of the inferred trend itself flipped depending on selection rule.

Claim: Gutiérrez-Fandiño et al.'s persistence-diagram-distance method does not subsample the network at all — it uses every neuron and connection at every training step

Gutiérrez-Fandiño, Pérez-Fernández, Armengol-Estapé & Villegas (2021) build their simplicial complex from the complete weighted, directed graph of the network at each training batch: "For every training state, neural network connections are considered as directed and weighted edges between neurons, represented by graph nodes." Their own stated limitation is explicit that no node- or weight-level subsampling occurs anywhere in the pipeline: "we do not simplify the graph representation of the neural networks (we keep every single neuron and connections) and we do not approximate any computation." The paper's stated cost of this choice is compute, not sampling risk — 7 days on an HPC machine — which the authors flag as the method's "main limitation," explicitly a scalability problem rather than a selection-bias one.

Because there is no node-selection step in this method's design, a hub-selection-style artifact (which specific nodes get sampled shifting the sign of the signal) has no point of entry — the method reads the whole network every time. This stands in direct contrast to claim-hub-selection-artifact-can-reverse-network-breakpoint-signal, where the vulnerability exists precisely because the Byzantine study's trade network is too large to use in full and must be subsampled.

Claim: The general TDA subsampling-stability literature bounds instability from noise, density, and outliers — not from selection by a structural covariate

Two Tier-1 papers anchor the statistical theory behind subsampling persistence diagrams, and neither addresses degree-like structural selection. Chazal, Fasy, Lecci, Michel, Rinaldo & Wasserman (2014/ICML 2015) prove subsampled persistence-diagram/landscape estimators are "stable with respect to perturbations of the underlying measure" and that their estimator "is robust to the presence of outliers" — their subsamples are drawn i.i.d. at random from the point cloud, and stability is proven against distributional noise, not against a choice of which systematic subset (e.g., top-degree nodes) to sample. Stolz (2023, JMLR) surveys the two standard landmark-selection rules used to subsample point clouds for PH — "random selection tends to favour dense areas of the data while the maxmin algorithm is very sensitive to noise" — and proposes a topology-preserving alternative, explicitly framed around noise and density bias, with "robustness to outliers" as the paper's own evaluation criterion. 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." Nothing in this literature proves PH-based estimators are immune to a hub-selection-style artifact; it simply has not been the question these stability results were built to answer.

Claim: No hub-selection-style artifact is documented, tested, or ruled out in either founding NN-PH generalization paper — the vulnerability is neither confirmed nor formally excluded

Reading both papers' full method, ablation, and limitations sections directly turned up no experiment resembling the Byzantine study's robustness check (comparing a structural selection rule against a coverage-preserving alternative and showing the inferred trend's sign changes). Birdal et al. test sensitivity to sample size (random subsampling at varying n) but not to selection rule; Gutiérrez-Fandiño et al. avoid the question by not subsampling at all. [unverified — could not confirm or deny the core question for the field as a whole after this search]: this capture establishes that the specific vulnerability is absent by design from these two papers' methods (one has no selection step; the other's only sampling axis is random and produces a different, non-sign-reversing bias), and that the general TDA subsampling literature addresses a different threat model (noise/density/outliers, not structural selection). It does not establish that no PH-based NN diagnostic anywhere is vulnerable — that would require surveying diagnostics that do subsample by a structural rule (see Further leads), which this session did not locate.

Further leads

• Rieck et al. 2019 "Neural Persistence" (cited as [RTB+19] in Birdal et al.'s related work) computes topology directly on trained layer weights rather than a trajectory sample — worth checking whether its construction has a weight-magnitude selection step that could carry a hub-selection-style risk.
• Corneanu et al. 2020 (CVPR), "Computing the Testing Error Without a Testing Set" — both NN-PH papers in this capture criticize its correlation-based metric-space construction as "numerically brittle"/poorly scaling; worth checking if that brittleness is itself a selection-sensitivity issue.
• Chowdhury et al. 2019, "Path homologies of deep feedforward networks" (directed-homology representation) — cited as related work by Gutiérrez-Fandiño et al.; sampling method not checked this session.
• The Byzantine study's own fix — a minimum-coverage sampling rule preserving regional representation ratios — has no documented analog in NN-PH diagnostics found this session; a dedicated search for "adversarial subsampling" or "checkpoint selection bias" in PH-based generalization literature could close this gap.

Entity candidates

• Tolga Birdal — person — first author of the PHD/generalization-bound paper anchoring one side of the vault's PH bridge
• Umut Şimşekli — person — co-author on Birdal et al.; also connects to heavy-tailed-SGD generalization literature already touched by the vault
• Asier Gutiérrez-Fandiño — person — first author of the validation-set-free PH-diagram-distance paper
• Frédéric Chazal — person — leading statistical-TDA theorist; first author of the foundational subsampling-stability paper
• Bernadette J. Stolz — person — author of the outlier-robust landmark-selection paper; recent (2023) entrant to the subsampling-for-PH literature
• persistent homology dimension (PHD) — concept — the specific estimator Birdal et al. define, distinct from generic persistent homology
• landmark selection (topological data analysis) — concept — the general open problem of choosing which points to subsample for PH; directly the axis the hub-selection question probes
• Neural Persistence (Rieck et al. 2019) — term — a third PH-based network diagnostic, computed on trained weights rather than a trajectory, not yet examined in the vault