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Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack

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Pith's one-line read A color-constrained Gromov-Hausdorff distance makes the six-pack of persistence diagrams stable.

desk verdict A correct and genuinely useful stability theorem for chromatic persistence; two minor issues don't threaten the core. read the letter →

arxiv 2507.17994 v1 pith:2DPELB44 submitted 2025-07-24 math.MG cs.CG

Ondřej Draganov , Sophie Rosenmeier , Nicolò Zava This is my paper · ORCID
classification math.MGcs.CG MSC 53C2355N31
keywords chromaticmetricpairsC-constrainedGromov-Hausdorffdistancesix-packofpersistencediagramsbottleneckstabilityambientČechfiltrationcoloredpointcloudsAlexandrovtopologyoncolors
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper defines a distance between chromatic metric pairs — metric spaces carrying a coloring on a distinguished subspace — and proves that a standard invariant called the six-pack of persistence diagrams is stable under this distance. The distance, the C-constrained Gromov-Hausdorff distance, only admits comparison maps that respect chosen color constraints, and reduces to the classical Gromov-Hausdorff distance when colors carry no information. The main theorem states that when two finite colored point sets are ε-close in this distance, with constraints taken from the maximal faces of the two simplicial complexes that define the six-pack, all six persistence diagrams are at most 2ε apart in bottleneck distance. This turns the six-pack into a certificate that can be compared across datasets with labeled subpopulations or components.

What carries the argument

The central object is the $C$-constrained Gromov-Hausdorff distance, defined as half the infimum over pairs $f:A_1\to A_2$, $g:A_2\to A_1$ of $\max\{\operatorname{dis} f, \operatorname{dis} g, \operatorname{codis}(f,g)\}$, where all maps are required to be $C$-constrained: for every $\sigma\in C$, $f(\chi_1^{-1}(\sigma))\subseteq \chi_2^{-1}(\sigma)$. The stability proof is carried by Lemmas 5.2–5.4: a $C(\Gamma)\cup C(\Lambda)$-constrained pair of maps with distortion and codistortion at most $2\varepsilon$ induces, for every radius $\delta$, simplicial maps between the $\Gamma$- and $\Lambda$-subcomplexes of the ambient Čech filtrations, and these maps form interleavings that commute up to contiguity. Homology converts these into genuine $2\varepsilon$-interleavings of the six persistence modules, and the algebraic stability theorem turns those into bottleneck-distance bounds on the persistence diagrams.

What would settle it

Take two finite colored point sets in the plane with $\Lambda = \{\{0\}\}$ and $\Gamma = \{\{0\},\{1\},\{0,1\}\}$, compute $d^{C}_{GH}$ for $C=\{\{0\},\{0,1\}\}$ by the pair-of-maps definition, and compute the six persistence diagrams on a fine discretization, as in Example 5.6; the theorem is false if any instance gives $d_B$ larger than $2\,d^{C}_{GH}$.

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Extended reading notes

Core claim

The paper's central claim is that the six-pack is stable with respect to the constrained distance: for finite colored subsets $X_1 \subseteq A_1$ and $X_2 \subseteq A_2$, with $\Lambda \leq \Gamma$ finite-dimensional simplicial complexes on the color set and $C = C(\Gamma) \cup C(\Lambda)$ the union of their sets of maximal faces, each of the six persistence modules — domain, codomain, image, kernel, cokernel, and relative homology built from the inclusions $\check{C}^\Lambda(\chi;A) \hookrightarrow \check{C}^\Gamma(\chi;A)$ — satisfies $d_B(\operatorname{Dgm} U_1, \operatorname{Dgm} U_2) \leq 2\,d^{C}_{GH}((A_1,\chi_1),(A_2,\chi_2))$. The constraining family $C$ is not incidental: it is exactly what guarantees that a $C$-constrained map sends $\Gamma$-colored simplices of the Čech filtration to $\Gamma$-colored simplices at a controlled radius shift (Lemma 5.2).

Load-bearing premise

The whole bound rests on the existence, for nearly equal colored point sets, of two maps that respect exactly the chosen color groupings and have both distortion and codistortion below $2\varepsilon$; if the constraint set is looser or the point sets are infinite, the interleaving construction is not guaranteed and the bottleneck bound can fail.

Editorial extensions

If this is right

  • If two finite colored point sets are close in $d^{C}_{GH}$ with $C=C(\Gamma)\cup C(\Lambda)$, then all six persistence diagrams in the six-pack are close in bottleneck distance, with constant 2.
  • The ambient Čech persistence diagrams of two finite metric pairs are stable with respect to the induced Gromov-Hausdorff distance for metric pairs (Corollary 5.7).
  • The new distance inherits classical characterizations: it can be computed via correspondences or common embeddings, and under compactness it is attained and vanishes exactly for constrained isomorphisms (Theorem 4.9, Theorem 4.13, Corollary 4.20).
  • Weaker constraint sets yield smaller distances, so invariants that ignore some color information can be bounded more tightly by choosing weaker constraints (Remark 4.3(iv), Example 4.4).
  • The six-pack stability generalizes earlier stability results for metric pairs and for degree-0 image, kernel, and cokernel persistence with Vietoris-Rips filtrations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same interleaving-by-contiguity skeleton should transfer to other geometric complexes, such as Vietoris-Rips or Dowker filtrations, since the proof only needs the radius-shift and contiguity lemmas, not the spherical geometry of Čech balls.
  • The paper's bracketed conjecture that the persistence diagrams of $(X,\operatorname{rad}_A)$ and of the ambient Čech filtration coincide, if true, would remove the circumradius-attainment hypothesis from the tripod-based proof and extend Corollary 5.7 to all finite metric pairs.
  • For labeled point-cloud matching, this suggests a practical recipe: choose the weakest constraint set compatible with the labels an application cares about, compute $d^{C}_{GH}$, and use the six-pack bottleneck distances as certified lower bounds; Example 5.6 shows such lower bounds can be tight.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper introduces chromatic metric pairs (A, χ:X→N), C-constrained maps, and a C-constrained Gromov-Hausdorff distance d_C_GH defined by minimizing max{dis f, dis g, codis(f,g)} over C-constrained pairs of maps. It establishes basic properties of this distance, characterizations via C-constrained correspondences and via admissible metrics on disjoint unions, existence of optimal correspondences for C-compact pairs, and stable lower-bound invariants. The central result is Theorem 5.5: for finite X1 and X2 and finite-dimensional simplicial complexes Λ ≤ Γ on N, each of the six persistence modules (domain, codomain, image, kernel, cokernel, relative homology) satisfies dB(Dgm U1, Dgm U2) ≤ 2 d_{C(Γ)∪C(Λ)}_GH((A1, χ1), (A2, χ2)). Corollary 5.7 derives stability for ambient Čech persistence diagrams, and Section 5.3 gives an alternative tripod-based proof under an additional technical assumption.

Significance. The paper supplies a natural and genuinely new distance for colored metric data and proves the first stability theorem for the six-pack with respect to such a distance. The foundational development is careful: the Alexandrov-topology analysis of constraint sets in Section 3.1 is elegant and useful, the computations in Section 4.1 provide sharp worked examples, and the main theorem is a substantial extension of persistence stability to chromatic settings. The proof adapts Chowdhury–Mémoli interleaving techniques and the algebraic stability theorem in a coherent way, and the distance is defined independently of persistence, so there is no circularity. I concur with the stress-test assessment: the flagged issues in Proposition 4.12 and Section 5.3 are real but do not affect the central six-pack stability theorem. The main limitations are the finiteness assumption in Theorem 5.5 and the unresolved question in Section 5.3, both of which are stated explicitly in the manuscript.

minor comments (5)
  1. [§4.2, proof of Proposition 4.12] The displayed equality 'sup_σ sup_{(y,z)∈R_σ} d(y,z) = sup_{(y,z)∈R} d(y,z)' is false in general, because R also contains the arbitrary correspondence R′ and the supremum over R can be larger. The intended inequality d_C_H(Y,Z) ≤ max{d_H(Y,Z), sup_σ d_H(Y_σ,Z_σ)} can be obtained by choosing R′ and the R_σ with suprema close to the corresponding Hausdorff values, so the statement of the proposition is correct, but the proof as written needs this repair.
  2. [§5.3, bracketed question after Proposition 5.8] The bracketed sentence 'I suspect that even in the general setting the persistence diagrams of (X, rad_A) and Č(X;A) coincide, don’t they?' is an acknowledged open point inside the alternative proof. Since Corollary 5.7 follows directly from Theorem 5.5, this does not threaten the central claim, but the sentence should be resolved, removed, or converted into an explicit open question.
  3. [§5.3, proof of Proposition 5.8] In the proof, after choosing a2, the conclusion 'rad_{A2}(π1(σ)) ≤ rad_{A1}(π1(σ)) + dis R + ε' should refer to rad_{A2}(π2(σ)); the same notational slip occurs in the following displayed line.
  4. [§5.2, proof of Lemma 5.2] The sentence 'we define f_δ ... on the vertex set by setting f_δ(x) = f(x) for every vertex x of Γ' should read 'for every vertex x of Č_δ(X1; A1)' or 'of X1'; as written, it is confusing because Γ has vertices in N and is not the source complex of f_δ.
  5. [Throughout] There are several typos that should be corrected: 'Gromov-Hausdroff' in Section 2, 'C-cconstrained' in the caption of Figure 13, and 'a constraint set N∈C⊆P(N)' in Example 5.6 should read 'C⊆P(N) with N∈C'. None of these affects the mathematics.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the C-constrained Gromov-Hausdorff distance is defined independently, and Theorem 5.5 derives six-pack stability via external algebraic stability and contiguity arguments.

full rationale

The central derivation chain is self-contained against external benchmarks. dC_GH (Definition 4.1) is an independent infimum over C-constrained map pairs, and the six-pack modules (Section 5.1–5.2) are defined from the Cech filtrations and the inclusion Lambda <= Gamma. Theorem 5.5 does not define either quantity in terms of the other; it constructs 2-epsilon-interleavings in Lemma 5.3 (adapting [8, Proposition 15]) and then invokes the algebraic stability theorem [5]. The only author self-citations, [13]/[14], supply the six-pack construction as background and a parenthetical homotopy-equivalence remark, which is not used in the proof of Theorem 5.5. The alternative proof in Section 5.3 has an explicitly flagged gap: '[[I suspect that even in the general setting the persistence diagrams of (X, radA) and C(X;A) coincide, don't they?]]' and requires the circumradius infimum to be attained, but this route is not needed, since Corollary 5.7 follows from Theorem 5.5 by taking Gamma = {{0}} and Lambda = empty. The reversed displayed inequality in the proof of Proposition 4.12 is a proof/correctness typo (the statement is the reverse of the bound just established) and is not central to the stability theorem. No fitted parameter is relabeled as a prediction, and no uniqueness result is imported from the authors' own prior work. Hence the paper's derivation is not circular.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces a new distance function and its stability theory, but no new physical entities, free parameters, or fitted constants. The axioms are standard mathematical background plus the finiteness and circumradius assumptions that define the scope of the main theorems.

assumptions (5)
  • standard math ZFC set theory and standard point-set topology
    All proofs assume standard mathematical foundations, including the axiom of choice when selecting maps from correspondences in Theorem 4.9.
  • standard math Pointwise finite-dimensional persistence modules decompose into interval modules (Crawley-Boevey)
    Used in Section 5.1 to define persistence diagrams from persistence modules; cited as [12].
  • standard math Algebraic stability theorem for q-tame modules (Chazal, Cohen-Steiner, Glisse, Guibas, Oudot)
    Theorem 5.1, cited as [5], converts the constructed 2ε-interleavings into bottleneck distance bounds.
  • domain assumption Finiteness of X1 and X2 in the main stability theorem
    Theorem 5.5 assumes X1 and X2 are finite, ensuring the Cech filtrations are finite at each radius and the persistence modules are pointwise finite-dimensional. The paper does not extend the stability result to infinite colored subsets.
  • domain assumption The circumradius infimum in equation (8) is attained for every subset
    Section 5.3 uses this to identify the filtered space (X, rad_A) with the ambient Cech filtration in the alternative proof of Corollary 5.7. The paper leaves the general case open with a bracketed question.

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Pith. "Pith review of Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack." pith.science (2026). https://pith.science/paper/2DPELB44

@misc{pith2026250717994,
  author       = {Pith},
  title        = {Pith review of: Gromov-Hausdorff distance between chromatic metric pairs and stability of the six-pack},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DPELB44}},
  note         = {Machine review of arXiv:2507.17994}
}
read the original abstract

Chromatic metric pairs consist of a metric space and a coloring function partitioning a subset thereof into various colors. It is a natural extension of the notion of chromatic point sets studied in chromatic topological data analysis. A useful tool in the field is the six-pack, a collection of six persistence diagrams, summarizing homological information about how the colored subsets interact. We introduce a suitable generalization of the Gromov-Hausdorff distance to compare chromatic metric pairs. We show some basic properties and validate this definition by obtaining the stability of the six-pack with respect to that distance. We conclude by discussing its restriction to metric pairs and its role in the stability of the \v{C}ech persistence diagrams.

Figures

Figures reproduced from arXiv: 2507.17994 by the authors.

Figure 1
Figure 1. Hausdorff distance for two subsets of the plane: A, a circle, in cyan, and B, a collection of dots, in magenta. Since A is contained in the union of balls centered in the points of B with radius ε (right), and B lies inside the union of balls centered in points of A with the same radius (center), their Hausdorff distance is at most ε. The Hausdorff distance provides another, more geometrical, characterization of the… view at source ↗
Figure 2
Figure 2. On the left-hand side, an example of a chromatic metric pair (A, χ) and a subset Y of A. On the right-hand side, we highlight the chromatic metric pair structure on Y induced by (A, χ). Remark 3.3. Even though by definition we do not require the maps between chromatic metric pairs to be pair maps, in the rest of the paper, we always require N to be part of any constraint set C, which has the same effect. We could al… view at source ↗
Figure 3
Figure 3. A representation of the four chromatic metric spaces described in Example 3.7. (in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: A representation of the two chromatic metric spaces defined in Remark 4.3(i). Set C = {{0}, N}. Then, d C GH(χ1, χ2) > 0, but dGH(X1, X2) = dGH(χ −1 1 ({0}), χ−1 2 ({0})) = 0 since both pairs are isometric. (ii) We have discussed in Remark 3.5 that metric spaces and me…
Figure 5
Figure 5. Figure 5: A representation of the four chromatic metric pairs defined in Example 4.4(ii). The points colored in 0 are represented in magenta, and in cyan, those colored in 1. First, we can immediately note that d CD GH(χ1, χ2) = d CD GH(χ1, χ3) = d CD GH(χ1, χ4) = d CD GH(χ2, χ4…
Figure 6
Figure 6. Figure 6: The chromatic metric pair (R2 , χ) as defined in Example 4.6. The colors 0 and 1 are represented in magenta and cyan, respectively. Example 4.6. Let us discuss a quick example to demonstrate the different invariants defined above. Take the chromatic metric pair (R 2 , …
Figure 7
Figure 7. Figure 7: A representation of the chromatic metric spaces χ1 and χ2 defined in Example 4.8(i). Consider the discrete constraint set CD. In this case, it is easy to see that χ −1 1 (σ) is isometric to χ −1 2 (σ) for every σ ∈ CD, and so the usual Gromov-Hausdorff distance does no…
Figure 8
Figure 8. Figure 8: A representation of the chromatic metric pairs χ1 and χε defined in Example 4.8(ii). the CD-constrained Gromov-Hausdorff distance between χ1 and χε. Clearly, χ −1 1 (0) and χ −1 ε (0), and χ −1 1 (1) and χ −1 ε (1) are pairwise isometric. Define two maps, fε : N ∪ {−1}…
Figure 9
Figure 9. Figure 9: A representation of the pairs of maps used to show upper bounds in Example 4.8(ii). On the left-hand side, we represent the maps fε (dashed) pointing downwards and gε (dotted) pointing upwards. On the right-hand side, we show the CD-constrained bijection h. We intend t…
Figure 10
Figure 10. Figure 10: A modification of the chromatic metric spaces provided in Example 4.8(ii). The two underlying metric spaces of χ ′ 1 and χ ′ ε are the subspaces Z and N ∪ (−N − ε) = N ∪ {−n − ε | n ∈ N} of the real line, respectively. The coloring functions satisfy χ ′ 1 : N 7→ 0, χ …
Figure 11
Figure 11. Figure 11: A representation of the example constructed in Remark 4.10. The three colors 0, 1, 2 are represented in cyan, violet and magenta, respectively. Since both R|{a,b}×{x} and R|{b,c}×{y} are correspondences, R is a C-constrained correspondence. However, if a map sends the…
Figure 12
Figure 12. Figure 12: A representation of the two chromatic metric pairs described in Example 5.6. The colors 0 and 1 are represented in magenta and cyan, respectively. (CˇΓ(χi ; Xi), CˇΛ(χi ; Xi)) for i = 1, 2 2 . Then, Dgm KerΛ≤Γ 0 (χ1; A1) has a single feature (0, r), whereas Dgm KerΛ≤Γ…
Figure 13
Figure 13. Figure 13: A representation of the C-cconstrained maps f (on the left-hand side) and g (on the right-hand side) constructed in Example 5.6 to provide an upper bound to the C-constrained Gromov-Hausdorff distance between χ1 and χ2. 5.3 Stability of the ambient Cech persistence di…

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