REVIEW 4 major objections 4 minor 7 references
The Stability of Persistence Diagrams Under Non-Uniform Scaling
T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash
Pith's one-line read This paper claims a quantitative stability bound for persistence diagrams under coordinate-wise scaling: the bottleneck distance is at most $\frac{1}{2}(s_{\max}-s_{\min})\operatorname{diam}(X)$, with extensions to higher homology…
desk verdict The main bound is false: it misquotes the stability theorem, and the simplest two-point example already violates it. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the coordinate-wise scaling map $S$ together with the bottleneck distance $d_B$ between persistence diagrams. The argument's work is done by squeezing every pairwise distance between $s_{\min}d_X(p,q)$ and $s_{\max}d_X(p,q)$, which bounds the perturbation of every Vietoris-Rips or \v{C}ech filtration parameter by $(s_{\max}-s_{\min})\operatorname{diam}(X)$; a stability theorem is then invoked to convert that filtration perturbation into a bottleneck shift of half its size. The auxiliary quantity $s_{\rm avg}$, the root-mean-square of the scaling factors, supplies the lower bound, and the same squeezing is reapplied to $k$-simplices and to products of factors in the iterative setting.
What would settle it
Take $X=\{(0,0),(1,0)\}$ with scaling factors $s_1=2$ and $s_2=1$. The original Vietoris-Rips zero-dimensional diagram has one bar $[0,1)$; the scaled diagram has one bar $[0,2)$. The bottleneck distance is $|1-2|=1$, while the claimed upper bound $\frac{1}{2}(s_{\max}-s_{\min})\operatorname{diam}(X)=\frac{1}{2}(2-1)\cdot 1=\frac{1}{2}$ is violated under the paper's own definition of $d_B$ with matching to the diagonal allowed.
Extended reading notes
Core claim
On the paper's own terms, the central discovery is Lemma 3.1: for a finite metric space $X\subset\mathbb{R}^n$ with Euclidean distance and a coordinate-wise scaling $S(x_1,\ldots,x_n)=(s_1x_1,\ldots,s_nx_n)$, the bottleneck distance $d_B(D,D_S)$ between the persistence diagram of $X$ and that of $S(X)$ satisfies $$\frac{1}{2}(s_{\rm avg}-s_{\min})\operatorname{diam}(X)\le d_B(D,D_S)\le\frac{1}{2}(s_{\max}-s_{\min})\operatorname{diam}(X),$$ where $s_{\rm avg}=\sqrt{\frac{1}{n}\sum_i s_i^2}$. The same upper bound is extended to each homology degree $k$, to the $p$-Wasserstein distance, to a composition of $m$ successive scalings with $s_{\max}$ and $s_{\min}$ replaced by the products $\prod_j s^{(j)}_{\max}$ and $\prod_j s^{(j)}_{\min}$, and to expectations over random scaling factors.
Load-bearing premise
The load-bearing premise is that changing every pairwise distance in a dataset by at most some amount moves the persistence diagram by at most half that amount in the bottleneck metric; if the true factor is the full amount instead of half, all the stated constants double.
Editorial extensions
If this is right
- If the bound holds, then feature-by-feature normalization of data changes the persistent homology output by an amount controlled solely by the spread of the scaling factors and the original diameter, not by the number of points or the ambient dimension.
- The dimension-dependent version implies that cycles in every homology degree move by at most the same linear bound, so higher-dimensional topological features are not disproportionately destabilized by anisotropic scaling beyond the larger diameters they may have.
- The iterative version implies that composing multiple scalings compounds as the product of the maxima minus the product of the minima, so alternating stretches and compressions do not cancel even when their factors mirror each other.
- The Wasserstein statement implies that both the worst-case matching and the average transport cost between diagrams are bounded by the same linear expression in the scaling range and dataset diameter.
- The probabilistic version gives an expectation bound when scaling factors are random, so preprocessing noise can be budgeted in expectation rather than only in the worst case.
Reading between the lines
- A reader should verify which stability convention is being used: the standard stability theorem bounds $d_B$ by the perturbation itself, not by half of it. If the half factor is dropped, the main inequality becomes $d_B(D,D_S)\le (s_{\max}-s_{\min})\operatorname{diam}(X)$, which still gives a linear-in-spread bound.
- As printed, the proof of Lemma 3.1 cites Lemma 3.1 in its own first paragraph to assert the distance bounds; the intended step is the elementary $s_{\min}d_X\le d_S\le s_{\max}d_X$, which is enough for the rest of the argument.
- A synthetic check on random point clouds with known diameter and prescribed scaling factors would reveal whether the observed bottleneck distance tracks $\frac{1}{2}(s_{\max}-s_{\min})\operatorname{diam}(X)$ or the full $(s_{\max}-s_{\min})\operatorname{diam}(X)$, separating the paper's constant from the standard stability constant.
- The compounding formula for iterative scalings suggests a testable preprocessing warning: even small alternating stretches and compressions multiply into a nonzero scaling spread, so repeated normalization steps can accumulate distortion rather than cancel.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims that for a finite metric space X ⊂ R^n with Euclidean distance and a coordinate-wise scaling S with factors s_i, the bottleneck distance between the persistence diagram D of X and the diagram D_S of S(X) satisfies 1/2 (s_avg − s_min) · diam(X) ≤ d_B(D, D_S) ≤ 1/2 (s_max − s_min) · diam(X), where s_avg is the root-mean-square scaling factor. It further extends this bound to higher homology, the Wasserstein distance, iterative scalings, and random scalings, and illustrates the results with case studies. The proof strategy is to bound the perturbation of Rips filtration values by (s_max − s_min) · diam(X) and then invoke a stability theorem with an alleged factor of 1/2.
Significance. The question addressed is natural: anisotropic scaling is common in applications, and a quantitative bound on persistence diagram distortion under non-uniform scaling would be useful. However, the central bound is false, as shown by a two-point counterexample, and the proof rests on an incorrect statement of the stability theorem. The paper contains no machine-checked proofs or numerical experiments that could independently confirm the claims; the case studies merely substitute values into the asserted formula. Because the main theorem fails, all downstream extensions are unsupported.
major comments (4)
- [Section 3.1, Lemma 3.1] The upper bound in Lemma 3.1 is false. Let X = {(0,0),(1,0)} ⊂ R^2 and S(x,y) = (2x, y), so s_max = 2, s_min = 1, and diam(X) = 1. The claimed bound gives d_B(D, D_S) ≤ 1/2 · (2 − 1) · 1 = 0.5. For the Vietoris–Rips filtration, the 0-dimensional diagrams are D = {(0,1)} and D_S = {(0,2)}, whose bottleneck distance is 1, because the optimal matching of the single off-diagonal point pairs has cost |2 − 1| = 1. This contradicts the asserted inequality.
- [Section 3.2, proof of Theorem 3.2; also used in Theorems 3.3–3.5] The proof invokes the statement 'By the stability theorem for persistence diagrams, the bottleneck distance is bounded by half the maximum perturbation in ε'. This is not the standard stability theorem. The standard theorem bounds the bottleneck distance by the interleaving/perturbation parameter ε itself, not by ε/2. In the Rips setting, a distance perturbation of δ shifts filtration values by δ, so the correct constant is 1, not 1/2. This mistaken half-factor is exactly what produces the false bound in the counterexample above, and since it is used in every subsequent theorem, those theorems are all unsupported.
- [Section 3.1, lower bound in Lemma 3.1] The lower bound is asserted without proof and is also false. Take X = {(0,0),(10,0),(5,1)} ⊂ R^2 and scaling factors s = (1,2). Then s_min = 1, s_avg = sqrt((1^2+2^2)/2) = sqrt(5/2), and diam(X) = 10, so the claimed lower bound is 1/2(sqrt(5/2) − 1) · 10 ≈ 2.905. The original H_0 diagram has two off-diagonal points at (0,√26); the scaled diagram has two points at (0,√29). Hence d_B(D, D_S) = √29 − √26 ≈ 0.286, which is far below the asserted lower bound.
- [Section 3.4, Theorem 3.4] The proof of the Wasserstein bound uses the inequality W_p(D_1,D_2) ≤ d_B(D_1,D_2), which is generally false. For two diagrams each with two off-diagonal points whose optimal matching costs are both 1, the p-Wasserstein distance is (1^p + 1^p)^{1/p} > 1 for finite p, while the bottleneck distance is 1. Thus the Wasserstein bound does not follow from the bottleneck bound even if the latter were corrected.
minor comments (4)
- [Section 3.3] The definitions s_total_min = ∏_j s_(j)_min and s_total_max = ∏_j s_(j)_max are not the cumulative per-coordinate extrema; the actual cumulative minimum is min_i ∏_j s_(j,i). The stated equalities are therefore wrong, although replacing them by the (looser) product bounds may still give a conservative upper bound.
- [Section 3.5] The proof contains a paragraph beginning 'The bound Wp(D, DS) ≤ ... has several important implications' that appears to be copied from Section 3.4 and is unrelated to the probabilistic expectation argument.
- [Section 4.3 and Section 4.4] Section 4.3 refers to 'Theorem 5.1' and Section 4.4 refers to 'Proposition 4.1', but neither exists in the manuscript; the references should be to the relevant lemma or theorem.
- [Section 3.4] The main result is stated as Lemma 3.1, but Section 3.4 cites it as 'Theorem 3.1'; this inconsistency should be fixed.
Circularity Check
No circularity: the paper derives its bounds from an external stability theorem and internal reuse of its own Lemma 3.1, which is normal proof structure, not circularity.
full rationale
I found no step in which a claimed prediction is equivalent by construction to an input or fitted parameter. Lemma 3.1 bounds d_B(D,D_S) by 1/2 (s_max - s_min) diam(X); the proof reasons from distance bounds s_min d_X <= d_S <= s_max d_X and then invokes 'the stability theorem for persistence diagrams' to halve the perturbation. That step is a mistaken application of an external theorem from Cohen-Steiner--Edelsbrunner--Harer, not a redefinition or self-citation, so it is a correctness error rather than circularity. The lower bound involving s_avg is asserted rather than derived, but an unsupported assertion is not a circular reduction. Theorems 3.2--3.5 all cite Lemma 3.1, but building later results on an earlier lemma is standard derivation-chain structure, not circularity; no parameter is fitted to data and no quantity is renamed as a prediction. The Wasserstein claim W_p <= d_B in Theorem 3.4 is false in general, and the half-factor stability statement in Section 3.2 is mathematically wrong, but both are external mathematical errors, not self-referential constructions. Accordingly the circularity score is 0.
Assumptions & free parameters
assumptions (4)
- ad hoc to paper The bottleneck distance between persistence diagrams is bounded by half the maximum perturbation of the filtration parameter
- ad hoc to paper Uniform scaling leaves the persistence diagram unchanged, so the bottleneck distance is zero when s_min = s_max
- ad hoc to paper The cumulative scaling factors after iterative transformations are the products of the per-step minima and maxima
- ad hoc to paper The expected minimum and maximum of n i.i.d. Uniform(a,b) variables are a and b
Cite this review
Pith. "Pith review of The Stability of Persistence Diagrams Under Non-Uniform Scaling." pith.science (2026). https://pith.science/paper/PXLFZWKS
@misc{pith2026241116126,
author = {Pith},
title = {Pith review of: The Stability of Persistence Diagrams Under Non-Uniform Scaling},
year = {2026},
howpublished = {\url{https://pith.science/paper/PXLFZWKS}},
note = {Machine review of arXiv:2411.16126}
}
abstract
We investigate the stability of persistence diagrams \( D \) under non-uniform scaling transformations \( S \) in \( \mathbb{R}^n \). Given a finite metric space \( X \subset \mathbb{R}^n \) with Euclidean distance \( d_X \), and scaling factors \( s_1, s_2, \ldots, s_n > 0 \) applied to each coordinate, we derive explicit bounds on the bottleneck distance \( d_B(D, D_S) \) between the persistence diagrams of \( X \) and its scaled version \( S(X) \). Specifically, we show that \[ d_B(D, D_S) \leq \frac{1}{2} (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), \] where \( s_{\min} \) and \( s_{\max} \) are the smallest and largest scaling factors, respectively, and \( \operatorname{diam}(X) \) is the diameter of \( X \). We extend this analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our results provide a framework for quantifying the effects of non-uniform scaling on persistence diagrams.
Reference graph
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Reviewed August 12, 2026 · model on record in the stance chip above.
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