On the Stability of the Euler Characteristic Transform for a Perturbed Embedding

Elena Xinyi Wang; Elizabeth Munch; Jasmine George; Messiah Ridgley II; Oscar Lledo Osborn

REVIEW 2 major objections 4 minor 24 references

On the Stability of the Euler Characteristic Transform for a Perturbed Embedding

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For two embeddings of the same simplicial complex, the Euler Characteristic Transform distance is at most a constant times the total vertex displacement.

desk verdict The ECT Lipschitz bound is a clean, correct composition of existing theorems, but the SELECT bound is proven only for piecewise-constant fields and the paper's attempt to extend it to PL functions rests on a false homotopy-retraction claim. read the letter →

arxiv 2506.19991 v1 pith:F3MJRZHM submitted 2025-06-24 cs.CG

Jasmine George , Oscar Lledo Osborn , Elizabeth Munch , Messiah Ridgley II , Elena Xinyi Wang This is my paper · ORCID

classification cs.CG MSC 55N3168U05

keywords EulercharacteristictransformSELECTsimplicialcomplexstabilityWassersteindistancepersistencediagramtopologicaldataanalysispiecewiselinearfield

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

In plain terms, the paper proves that two quantitative shape descriptors from topological data analysis are stable under small changes in the geometric embedding of a fixed simplicial complex. For the Euler Characteristic Transform, the result is an explicit bound: the distance between the transforms of two embeddings $f$ and $g$ of the same abstract complex $K$, measured by integrating the $L^1$ difference of their Euler characteristic curves over all viewing directions, is at most $2\,C_K C_d \sum_{v\in V(K)}\|f(v)-g(v)\|_2$. For the Super Lifted Euler Characteristic Transform, defined for scalar fields on shapes, a similar bound holds with an extra factor equal to the range of the field, when the field is piecewise constant and simplex-wise monotone. This matters because the ECT is already used in shape classification and machine-learning pipelines, and a proven stability bound says that small vertex perturbations cannot arbitrarily distort the descriptor.

What carries the argument

The load-bearing machinery is a chain of Wasserstein comparisons. The paper builds the ECT distance as $\int_{S^{d-1}}\|\mathrm{ECC}_\nu(f(K))-\mathrm{ECC}_\nu(g(K))\|_1\,d\nu$, then bounds the integrand by $2W_{1,\infty}$ of the persistence diagrams of the two height filtrations, uses a known direction-integrated $W_{1,1}$ stability bound whose constants are $C_K$ (the maximum number of simplices incident to a vertex) and $C_d$ (a fixed sphere integral), and closes the chain with the inequality $W_{1,\infty}\le W_{1,1}$. For SELECT, the mechanism is that superlevel sets $K_t=\{x:\phi(x)\ge t\}$ of a simplex-wise monotone field are subcomplexes, so $\mathrm{SELECT}(\phi\circ f^{-1})(\nu,a,t)=\mathrm{ECC}_\nu(f(K_t),a)$, turning each fixed-$t$ slice into an ECT computation and leaving a single integral over $t\in[0,r_{\max}]$.

What would settle it

For the ECT bound, take the simplest nontrivial complex, a single edge in $\mathbb{R}^2$, fix embeddings $f$ and $g$ that differ by moving one endpoint a distance $\varepsilon$, and compute $d_{\mathrm{ECT}}$ exactly; the theorem fails if it exceeds $2C_KC_d\varepsilon$. For the SELECT bound, take a genuine piecewise-linear interpolation $\Phi$ of vertex weights on a triangle, move one vertex by $\varepsilon$, and compare $d_{\mathrm{SELECT}}(\Phi\circ f^{-1},\Phi\circ g^{-1})$ with $2r_{\max}C_KC_d\varepsilon$; exceeding that value would show the asserted replacement of PL fields by piecewise-constant fields does not preserve the bound.

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the ECT and SELECT are Lipschitz functions of the vertex positions of a triangulation. Theorem 4.1 states that for two embeddings $f,g$ of the same abstract simplicial complex $K$ into $\mathbb{R}^d$, the distance $d_{\mathrm{ECT}}(\mathrm{ECT}(f(K)),\mathrm{ECT}(g(K)))$, defined by integrating the $L^1$ distance of Euler characteristic curves over all directions $\nu\in S^{d-1}$, satisfies $d_{\mathrm{ECT}}\le 2C_KC_d\sum_{v\in V(K)}\|f(v)-g(v)\|_2$, where $C_K$ counts the maximum number of simplices containing any vertex and $C_d$ is a constant depending only on $d$. The proof chains three existing inequalities: an Euler-characteristic-curve difference is bounded by a Wasserstein distance between persistence diagrams, that Wasserstein distance integrated over all directions is bounded by a constant times total vertex displacement, and the two Wasserstein metrics are comparable. Theorem 5.3 gives an analogous bound for SELECT: for a simplex-wise monotone vertex function $\phi$, the SELECT distance between $\mathrm{SELECT}(\phi\circ f^{-1})$ and $\mathrm{SELECT}(\phi\circ g^{-1})$ is at most $2r_{\max}C_KC_d\sum_{v}\|f(v)-g(v)\|_2$, by recognizing that SELECT at threshold $t$ is exactly the ECT of the superlevel subcomplex $K_t$ and integrating the ECT bound over $t$ up to $r_{\max}$.

Load-bearing premise

The SELECT result is proved only for step-like fields that are constant on each simplex; the paper's argument that this also covers ordinary piecewise-linear interpolation depends on an asserted homotopy equivalence between the PL superlevel sets and the simplicial superlevel sets, and the paper lists the PL case itself as future work.

Editorial extensions

If this is right

If Theorem 4.1 is correct, the ECT is a Lipschitz map from the space of embeddings of a fixed abstract complex (metrized by total vertex displacement) to the space of transforms with the direction-integrated $L^1$ distance.
Small geometric perturbation of a shape whose triangulation is unchanged changes the ECT by at most a constant times the sum of vertex movements, with the constant computable from the complex and the dimension.
The SELECT bound extends the same guarantee to scalar fields: for step-like fields satisfying the simplex-wise monotone condition, moving the embedding changes SELECT by at most a constant times the total vertex displacement times the field range $r_{\max}$.
Because all constants are explicit, a practitioner can predict the worst-case descriptor drift from a known mesh deformation before recomputing the transform.

Reading between the lines

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

The same chain of inequalities would likely prove stability for any transform that controls its per-direction output by a Wasserstein distance between persistence diagrams; the authors do not state this general template.
The ECT bound implies a statistical continuity the paper leaves implicit: if vertex positions are estimated with small registration error, empirical ECT-based summaries change by at most a fixed multiple of that error, so confidence statements about shape descriptors inherit continuity in the mesh.
The factor $r_{\max}$ in the SELECT bound is plausibly loose, since $K_t$ shrinks as $t$ grows and $C_{K_t}$ decreases; a sharper bound integrating only over the actual spread of vertex values might be strictly smaller.
A closed-form check on a single edge or triangle would calibrate how much slack the constants $C_K$ and $C_d$ carry, since the paper gives no optimality example.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the ECT half is a correct, clean composition of known inequalities, and the explicit Lipschitz constant is worth having. The SELECT half is a real extension but is proven only for piecewise-constant, simplex-wise monotone fields. The paragraph that tries to bridge to PL functions asserts a homotopy equivalence that does not survive the halfspace truncation SELECT evaluates; the stress-test counterexample with a single triangle is correct. So the SELECT claim should be explicitly scoped to piecewise-constant fields, with PL interpolation left as future work—which the Discussion already lists.

What is actually new and good: Theorem 4.1 states an explicit bound with constants C_K and C_d, chaining Dlotko-Gurnari, Skraba-Turner, and Turner with all citations present and no hidden parameters. The proof is transparent. The SELECT bound's reduction of d_SELECT to an integral of d_ECT over t is neat, and the steps C_{K_t} ≤ C_K and V(K_t) ⊆ V(K) are correct. The writing is clear, and the figures borrowed from Munch are well chosen.

Soft spots, in order of importance. First, the PL bridge. The paragraph beginning "However, since this idea is closely related..." claims that Φ^{-1}[t,∞) retracts to K_t and therefore the piecewise-constant function can be used instead. The homotopy equivalence of the full superlevel sets is true, but SELECT evaluates {x : ⟨x,ν⟩ ≤ a, Φ(x) ≥ t}, and truncation by the halfspace breaks the equivalence. The concrete counterexample in the stress-test note is exactly right: the truncated subtriangle has Euler characteristic 1 while the piecewise-constant replacement gives 0. So Theorem 5.3 is not established for PL fields. This is a scope problem, not a fatal one, because the proof is sound for the piecewise-constant fields it actually uses, and the Discussion already names PL interpolation as future work. The authors should delete or repair that paragraph and state the theorem for simplex-wise monotone fields only.

Second, the proof of Lemma 3.6 is omitted. Minor; it really is almost identical to Lemma 3.5. Third, the proof of Theorem 4.1 cites "Lemma 3.2," which does not exist; it should cite Theorem 3.3 (Turner's W_{p,∞} ≤ W_{p,p}). Minor typo, but confusing.

Citation pattern: no self-citation issues. Munch [15] is used only as a survey and figure source, which is appropriate. The external theorems are all properly credited.

Who this is for: TDA practitioners who want stability guarantees for deformable objects with fixed triangulation, and anyone building ML pipelines on ECT or SELECT. The bound is not tight but explicit, which is what practice needs.

Recommendation: send it to peer review. It deserves a serious referee. I would accept after minor revision, with the SELECT scope fixed and the two smaller issues cleaned up.

Referee Report

2 major / 4 minor

Summary. The paper studies stability of the Euler characteristic transform (ECT) and its field-valued variant SELECT under perturbations of an embedding of a fixed abstract simplicial complex K into R^d. Theorem 4.1 bounds the integrated L1 distance between the ECTs of two embeddings by 2 C_K C_d Σ_v ||f(v)-g(v)||_2, chaining a bound of Dlotko and Gurnari with the Wasserstein stability theorem of Skraba and Turner. For SELECT, the paper assigns vertex values φ, forms superlevel subcomplexes K_t, and proves Theorem 5.3, an analogous bound with an extra factor 2 r_max obtained by integrating Theorem 4.1 over t. The paper also proves two metric lemmas and concludes with a discussion of limitations, including the fact that piecewise-linear interpolation is left to future work.

Significance. The ECT stability result is a clean and useful contribution: it provides an explicit Lipschitz-type bound with constants depending only on the complex and the dimension, and the proof is a transparent chain of published theorems. The SELECT result is also valuable for piecewise-constant, simplex-wise monotone fields, where Theorem 5.3 gives an explicit bound with a natural dependence on the field range. However, the paper's advertised scope is broader than what is proved: the bridge used to pass from PL interpolation to piecewise-constant superlevel sets is invalid for the halfspace-truncated sets that SELECT evaluates, so the SELECT contribution needs to be explicitly scoped before the claims in the abstract and Definition 5.2.

major comments (2)

[Section 5, paragraph beginning 'However, since this idea...'] This paragraph asserts that a PL interpolation Φ of a simplex-wise monotone function has Φ^{-1}[t,∞) homotopy equivalent to K_t and therefore 'we are justified in using the piecewise constant function instead.' The equivalence is not enough: SELECT evaluates χ({x: ⟨x,ν⟩≤a, Φ(x)≥t}), and homotopy equivalence of the untruncated superlevel set does not survive intersection with the halfspace {⟨x,ν⟩≤a}. Concretely, take K to be the single triangle with vertices (0,0), (2,0), (2,2), φ(v1)=φ(v2)=1, φ(v3)=2, ν=(1,0), a=1.5, t=1.5. Then K_t={v3}, so the piecewise-constant replacement gives ECCν(f(K_t),1.5)=χ(∅)=0. The PL interpolation is Φ=1+λ_{v3}; its superlevel set {Φ≥1.5} is a subtriangle near v3, and its intersection with {x≤1.5} is nonempty and convex, so SELECT(Φ)(ν,a,t)=1. Hence Theorem 5.3 is not proved for PL fields. The theorem should be explicitly restricted to piecewise-constant/simplex-wise monotone fields, and the retraction paragraph should be repaired or removed; the Discussion already correctly lists PL interpolation as future work.
[Theorem 5.3 and Definition 5.2] There is a domain mismatch in the statement of Theorem 5.3: SELECT is defined in Definition 5.2 only for PL functions, but φ∘f^{-1} is piecewise constant and generally discontinuous, so it is not an element of PL(R^d). In addition, f^{-1}(x) is not uniquely defined for points on shared faces of simplices, and the value of φbar can differ on a face and on a higher-dimensional coface. The theorem should define ef and eg directly from the superlevel subcomplexes, for example ef(ν,a,t)=ECCν(f(K_t),a), and describe the fields as piecewise-constant fields associated to φ rather than as SELECT of a PL function.

minor comments (4)

[Proof of Theorem 4.1] The reference to 'Lemma 3.2' should be to Theorem 3.3, which is the inequality W_{1,∞}≤W_{1,1} used in that proof.
[Section 5, definition of r_max] Since φ is defined on vertices, the quantity r_max should be defined as max_{σ∈K} min_{v∈σ} φ(v) (or as the maximum of the vertex values), rather than max_{σ∈K} φ(σ), to avoid a domain mismatch.
[Section 5, text near Definition 5.2] There are several typos: 'continued' should be 'continuous', 'monoanalogous' should be 'monotone', and the definition 'ef:S^{d-1}×R×R' is missing its codomain.
[Abstract and Section 6] After scoping Theorem 5.3, the phrases 'fields defined on embedded simplicial complexes' in the abstract and 'the use of a particular class of field-inducing functions' in the Discussion should be made precise by referring to piecewise-constant, simplex-wise monotone fields.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the ECT and SELECT bounds chain external published theorems and explicit constants, with no fitted inputs; the only self-citation is a non-load-bearing survey/figure source.

full rationale

Theorem 4.1 is not circular. Its proof chains Theorem 3.1 (Dlotko-Gurnari [6]), Theorem 3.2 (Skraba-Turner [19]), and Theorem 3.3 (Turner [21]) to relate the integrated L1 distance of ECCs to a Wasserstein bound depending on vertex displacement. These are external published results, and the constants C_K and C_d are explicit rather than fitted. Theorem 5.3 is derived from the paper's own Theorem 4.1 by rewriting the SELECT distance as an integral over t of ECT distances of the superlevel subcomplexes K_t; this is internal derivation, not circularity, and r_max, C_K, C_d are explicit constants with no parameter fitted to the quantity being predicted. The only author self-citation is Munch [15], used as a survey and figure source for Figures 2 and 3; it is not load-bearing mathematical support. The LECT/SELECT definitions are attributed to Kirveslahti-Mukherjee [11], which is not a self-citation of the current authors. The text's retraction argument for piecewise-linear fields ('we can retract Phi^{-1}[t,infinity) to K_t') is not shown to preserve the Euler characteristic of the halfspace-truncated intersections {<x,nu> <= a} that SELECT evaluates; however, this is a scope/correctness concern, not a self-referential reduction. The paper itself lists piecewise-linear interpolation as future work and explicitly scopes the SELECT theorem to simplex-wise monotone fields, so the honest limitation is stated rather than hidden. Accordingly, the derivation is self-contained against external benchmarks and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No numbers are fitted to data. The constants C_d, C_K, and r_max are computed from dimension, complex structure, and field range. The derivation leans on three external theorems and on domain restrictions that the authors state. The one ad hoc ingredient is the replacement of PL fields by piecewise-constant superlevel complexes, which is asserted via homotopy equivalence and is not proved for height-truncated slices.

assumptions (7)

domain assumption All simplicial complexes are finite and can be assumed geometric; tame sets are replaced by their triangulations via the Triangulation Theorem (Theorem 2.2).
Used throughout to define ECC and ECT on finite geometric complexes; standard in TDA but restricts scope.
domain assumption The two compared shapes are geometric embeddings f,g of the same abstract simplicial complex K with fixed vertex correspondence.
Central to both theorems; guarantees equal Euler characteristic and finite ECC L1 distance, and excludes cross-triangulation comparisons.
domain assumption SELECT fields are piecewise-constant functions φ∘f^{-1} induced by a vertex function φ:V(K)→R_{>0} with superlevel sets K_t = ar φ^{-1}[t,∞) forming subcomplexes.
Used to rewrite the SELECT distance as an integral of the ECT distance over t; genuine PL interpolation is not covered.
standard math Dlotko-Gurnari Proposition 2: ||ECC_F(K1)-ECC_G(K2)||_1 ≤ 2 W_{1,∞}(Dgm(F),Dgm(G)).
External theorem, cited as [6], used as the first inequality in the proof of Theorem 4.1.
standard math Skraba-Turner Theorem 5.5: the integral of W_{1,1} between height-filtration persistence diagrams is bounded by C_K C_d times the sum of vertex displacements.
External theorem, cited as [19], supplies the embedding-perturbation bound.
standard math Turner's Theorem 3.3: W_{1,∞} ≤ W_{1,1} for persistence diagrams.
External theorem, cited as [21], closes the chain; the proof text mistakenly calls it 'Lemma 3.2'.
ad hoc to paper For a piecewise-linear interpolation Φ, the superlevel set Φ^{-1}[t,∞) is homotopy equivalent to K_t and hence has the same Euler characteristic, so the piecewise-constant function can be used instead.
Asserted, not proved, and not shown to preserve Euler characteristics of the height-truncated sets that SELECT actually evaluates; this is the weakest premise.

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Pith. "Pith review of On the Stability of the Euler Characteristic Transform for a Perturbed Embedding." pith.science (2026). https://pith.science/paper/F3MJRZHM

@misc{pith2026250619991,
  author       = {Pith},
  title        = {Pith review of: On the Stability of the Euler Characteristic Transform for a Perturbed Embedding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F3MJRZHM}},
  note         = {Machine review of arXiv:2506.19991}
}

read the original abstract

The Euler Characteristic Transform (ECT) is a robust method for shape classification. It takes an embedded shape and, for each direction, computes a piecewise constant function representing the Euler Characteristic of the shape's sublevel sets, which are defined by the height function in that direction. It has applications in TDA inverse problems, such as shape reconstruction, and is also employed with machine learning methodologies. In this paper, we define a distance between the ECTs of two distinct geometric embeddings of the same abstract simplicial complex and provide an upper bound for this distance. The Super Lifted Euler Characteristic Transform (SELECT), a related construction, extends the ECT to scalar fields defined on shapes. We establish a similar distance bound for SELECT, specifically when applied to fields defined on embedded simplicial complexes.

Figures

Figures reproduced from arXiv: 2506.19991 by the authors.

**Figure 2.** A “scan” of a fern leaf along the 3π 4 direction and the associated ECC. Figure from [15]. Definition 2.6. For simplicial complexes K1 and K2, scanned in directions ν and ω respectively, the L1 distance between their Euler Characteristic Curves is given by ∥ECCν(K1) − ECCω(K2)∥1 := Z R |ECCν(K1, a) − ECCω(K2, a)| da. This distance is finite only if the simplicial complexes K1 and K2 have the same Euler Characterist… view at source ↗

**Figure 3.** On the left, we have a simplicial complex with simplex-wise function [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [15]

An invitation to the Euler characteristic transform.arXiv preprint arXiv:2310.10395, 2023

Elizabeth Munch. An invitation to the Euler characteristic transform.arXiv preprint arXiv:2310.10395, 2023

arXiv 2023

[2] [1]

Measuring hidden phenotype: quantifying the shape of barley seeds using the Euler characteristic transform.in silico Plants, 4(1):diab033, 2022

Erik J Am´ ezquita, Michelle Y Quigley, Tim Ophelders, Jacob B Landis, Daniel Koenig, Elizabeth Munch, and Daniel H Chitwood. Measuring hidden phenotype: quantifying the shape of barley seeds using the Euler characteristic transform.in silico Plants, 4(1):diab033, 2022

work page 2022

[3] [2]

Stability of persistence diagrams.Discrete & Computational Geometry, 37(1):103–120, Jan 2007

David Cohen-Steiner, Herbert Edelsbrunner, and John Harer. Stability of persistence diagrams.Discrete & Computational Geometry, 37(1):103–120, Jan 2007

work page 2007

[4] [3]

Chen, Sayan Mukherjee, and Ra´ ul Rabad´ an

Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, and Ra´ ul Rabad´ an. Predicting clinical outcomes in glioblastoma: An application of topological and functional data analysis.Journal of the American Statistical Association, 115(531):1139–1150, 2020. 14

work page 2020

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Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms.Transactions of the American Mathematical Society, Series B, 9(32):1006–1043, 2022

2022

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Cambridge University Press, 2022

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2022

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Euler characteristic curves and profiles: a stable shape invariant for big data problems.GigaScience, 12:giad094, 2023

Pawe l D lotko and Davide Gurnari. Euler characteristic curves and profiles: a stable shape invariant for big data problems.GigaScience, 12:giad094, 2023

work page 2023

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work page 2008