Triangle inequality for the pullback distance of verbose persistence barcodes
Fix a field $\mathbb{F}$. For a finite nonempty pseudometric space $(S, d)$, form the Vietoris–Rips filtration in which a simplex $\sigma$ enters at time $\operatorname{diam}_d(\sigma)$, and let $V_k(S, d)$ be its degree-$k$ verbose barcode over $\mathbb{F}$: the multiset of all pairs of the usual boundary-matrix persistence pairing, including those with equal birth and death times. For $k \ge 1$ all bars have finite endpoints. For finite multisets $A, B$ of endpoint pairs of the same size let $$m(A, B) = \min_{\pi : A \to B} \max_{a \in A} \|a - \pi(a)\|_\infty,$$ the minimum over bijections, repeated elements counting as separately labelled copies, with $m(\varnothing, \varnothing) = 0$ and $m(A, B) = \infty$ for different sizes; no diagonal points are added. For finite nonempty metric spaces $X, Y$ let $$D_k(X, Y) = \inf_{S,\ p : S \twoheadrightarrow X,\ q : S \twoheadrightarrow Y} m\bigl(V_k(S, p^* d_X), V_k(S, q^* d_Y)\bigr),$$ over finite nonempty sets $S$ and surjections $p, q$, where $p^* d_X(s, t) = d_X(p(s), p(t))$; distinct points of $S$ stay distinct even at pullback distance zero. Zhou and Mémoli showed that $D_k(X, Y)$ is at least the bottleneck distance between the ordinary degree-$k$ barcodes and at most twice the Gromov–Hausdorff distance, that $D_0$ satisfies the triangle inequality, and that the analogous pullback interleaving distance does not.
Problem. Does $$D_k(X, Z) \le D_k(X, Y) + D_k(Y, Z)$$ hold for every field $\mathbb{F}$, every integer $k \ge 1$ and all finite nonempty metric spaces $X, Y, Z$?
Source: Zhou and Mémoli (2024), Remark 1.1(2) and Remark 5.17, for the pullback bottleneck distance of their Definition 5.3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AT — Algebraic Topology
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, persistent homology, verbose barcodes, Vietoris–Rips complexes
References
@article{zhou2024,
author = {Zhou, Ling and M{\'e}moli, Facundo},
title = {Ephemeral persistence features and the stability of filtered chain complexes},
journal = {J. Comput. Geom.},
volume = {15},
number = {2},
year = {2024},
pages = {258--328},
doi = {10.20382/jocg.v15i2a8},
eprint = {2208.11770},
archivePrefix = {arXiv},
}
@misc{aim-verbose-persistence-pullback-triangle,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Triangle inequality for the pullback distance of verbose persistence barcodes}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/534-verbose-persistence-pullback-triangle.md},
note = {Entry 534 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
The fixed-degree pullback matching distance of verbose Vietoris--Rips barcodes fails the triangle inequality. In degree one, three four-point ultrametric spaces, with nonzero distances only $1$ and $2$, have pairwise distances $0,0,1$. More generally, in every degree $k\ge1$ and over every field, the same phenomenon occurs on three spaces of equal cardinality $k+3$. Every barcode involved is nonempty. A rank-counting argument …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, siavash-sadeghi-544/aim544_counterexample.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00735, "Triangle inequality for the pullback distance of verbose persistence barcodes", clopen.solutions.