Birdal et al.'s persistent-homology-dimension estimator subsamples training iterates uniformly at random, not by any structural covariate

Birdal, Lou, Guibas & Şimşekli's PHD estimator (claim-birdal-2021-persistent-homology-dimension-bounds-generalization) is computed by an algorithm whose own pseudocode names its sampling step explicitly: "Wn ← sample(W, n) // random sampling." The experimental section confirms the same procedure in prose — "For varying n, we randomly pick n of last iterates and compute Eα0" — repeatedly drawing subsets of the recorded training trajectory at random, never by a structural property such as weight magnitude, layer, or training-loss rank.

The paper does document a known estimator bias tied to sample size: in its synthetic ground-truth ablation, "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 literature). This is a magnitude bias that shrinks monotonically as sample size grows — a materially different failure mode from a sign-reversing artifact tied to which points are chosen, of the kind claim-hub-selection-artifact-can-reverse-network-breakpoint-signal documents in a degree-heterogeneous trade network.

Because PHD's only sampling axis is uniform-random-over-time, the hub-selection mechanism has no structural-covariate entry point to exploit here — see observation-hub-selection-artifact-absent-by-design-in-founding-ph-generalization-papers for the fuller synthesis across both founding NN-PH diagnostics, which answers question-tda-neural-net-sampling-artifact-risk for the two papers it named.