{"id":"b02c0350-3486-42fd-8bc1-459152f925e8","arxiv_id":"2411.16126","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The claimed stability bound d_B ≤ 1/2 (s_max - s_min) diam(X) is false; the paper's own setup gives a counterexample where the bound is off by a factor of two.","lead":"This paper states a bound on the bottleneck distance between persistence diagrams before and after a dataset is stretched by different amounts along different coordinate axes. A simple two-point counterexample shows the bound is false, so the central theorem does not hold.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central bound is false: for X={(0,0),(1,0)} with s=(2,1), d_B(D,D_S)=1 but Lemma 3.1's upper bound is 0.5; the proof relies on an incorrect half-factor stability theorem.","rationale":"The reader's strongest_claim and weakest_assumption correctly locate the failure: the proof invokes a nonexistent half-factor stability theorem. Standard persistence stability gives d_B ≤ ε for an ε-perturbation, and for the Rips filtration a distance change of δ is an δ-perturbation of filtration values. The two-point counterexample is within the paper's own framework (finite subset of R^n, Euclidean distance, coordinate-wise scaling s_i > 0) and directly falsifies Lemma 3.1's upper bound. Because the upper bound is the foundation for all main theorems, the paper's central claim is false. The lower bound is also unproved, and the Wasserstein section reverses the true metric inequality (in general W_p ≥ d_B, not ≤), but the decisive issue is the false upper bound. The reader's REJECT verdict is correct; no adjustment is needed.","tokens_in":10794,"tokens_out":6194,"duration_ms":56205,"concrete_test":"Compute the 0-dimensional Vietoris–Rips persistence diagrams for the two-point metric spaces with distance matrices [[0,1],[1,0]] and [[0,2],[2,0]] (the latter is S(X) for s = (2,1)). With the paper's own definition of bottleneck distance, the diagrams are {(0,1)} and {(0,2)}, so d_B = 1. Compare with the claimed bound of 1/2 (s_max − s_min) diam(X) = 0.5. A one-line check in Ripser or by hand confirms the violation; this settles that the half-factor stability statement in §3.2 is false.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the assertion, made in the proof of Theorem 3.2 (and implicitly in Lemma 3.1, Section 3.1), that '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: an ε-interleaving of persistence modules implies bottleneck distance at most ε, not ε/2. For the Rips filtration used here, a distance perturbation of δ shifts filtration values by δ, so the correct constant is 1 (never 1/2). The error is fatal: take X = {(0,0),(1,0)} in R² with s = (2,1). Then diam(X) = 1, s_max − s_min = 1, so the claimed upper bound is 1/2. The 0-dimensional Rips diagrams are D = {(0,1)} and D_S = {(0,2)}, giving d_B = 1, violating the bound. Thus the central theorem and all downstream extensions (higher homology, Wasserstein, iterative, probabilistic) rest on a false premise. The lower bound in Lemma 3.1 is also unsupported, but the false half-factor upper bound is the decisive failure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":11122,"tokens_out":8287,"duration_ms":75820,"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":[{"comment":"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":"Section 3.1, Lemma 3.1"},{"comment":"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":"Section 3.2, proof of Theorem 3.2; also used in Theorems 3.3–3.5"},{"comment":"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":"Section 3.1, lower bound in Lemma 3.1"},{"comment":"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.","section":"Section 3.4, Theorem 3.4"}],"minor_comments":[{"comment":"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":"Section 3.3"},{"comment":"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":"Section 3.5"},{"comment":"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":"Section 4.3 and Section 4.4"},{"comment":"The main result is stated as Lemma 3.1, but Section 3.4 cites it as 'Theorem 3.1'; this inconsistency should be fixed.","section":"Section 3.4"}],"recommendation":"reject","confidential_remarks":"The central theorem is disproved by a simple explicit example, and the proof uses a nonstandard stability constant that is not supported by the literature. This is not a local fix: correcting the constant would change the claimed bound, and the proposed lower bound also fails on a separate example. I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know upfront. The main theorem is false, and the reason is a misapplied stability theorem: the paper uses a half-factor that is not in the standard statement. The two-point counterexample is decisive. Also, the lower bound in Lemma 3.1 is asserted, not proved.\n\nThe paper does address a real question — how anisotropic scaling moves persistence diagrams — and the setup is reader-friendly. The observation that coordinate-wise scaling puts all distances between s_min d and s_max d is correct, and the idea of using stability to convert that into a diagram bound is the right instinct. I want to credit that: the problem is worth a clean statement, and the paper is organized around the right chain.\n\nThe soft spot is central. The proof of Lemma 3.1 says the stability theorem bounds bottleneck distance by half the maximum filtration perturbation. It doesn't. For an ε-interleaving the bound is ε. For X = {(0,0),(1,0)} and s = (2,1), the H0 diagrams become (0,1) and (0,2), so d_B = 1, while the lemma's bound is 0.5. Every extension — higher homology, Wasserstein, iterative, random — inherits the failure. The Wasserstein theorem reverses the true inequality (W_p ≥ d_B for the norms used). The probabilistic theorem takes E[s_max] = b and E[s_min] = a for Uniform(a,b), which is wrong unless n is degenerate.\n\nThe case studies don't rescue anything; they illustrate the same flawed bounds. A correct version would use constant 1, drop the unsupported lower bound, and redo the Wasserstein and probabilistic sections. As written, this is a desk reject. I wouldn't cite it, and I wouldn't bring it to reading group except as a caution about checking the stability theorem's constants.","headline":"The main bound is false: it misquotes the stability theorem, and the simplest two-point example already violates it.","tokens_in":11583,"tokens_out":6675,"would_cite":false,"duration_ms":63573,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["persistent homology","persistence diagrams","bottleneck distance","non-uniform scaling","anisotropic scaling","stability theorem","Wasserstein distance","topological data analysis"],"falsifier":"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.","tokens_in":10567,"feed_emoji":"📐","tokens_out":10279,"duration_ms":89706,"temperature":0.7,"pith_summary":"Non-uniform scaling, multiplying each coordinate of a point cloud by its own positive factor, is common in image processing and feature normalization, yet its effect on persistent homology has not been quantified in this form. This paper tries to show that the bottleneck distance between the original and scaled persistence diagrams is bounded by $\\frac{1}{2}(s_{\\max}-s_{\\min})\\operatorname{diam}(X)$, where $s_{\\max}$ and $s_{\\min}$ are the largest and smallest scaling factors and $\\operatorname{diam}(X)$ is the dataset diameter. It also argues for a matching lower bound using the root-mean-square average scaling factor, and extends the upper bound to every homology degree, to Wasserstein distances, to compositions of several scalings, and to expected values under random scaling factors. If correct, this gives practitioners a simple formula for how much anisotropic rescaling can move the topological summary of their data, with distortion growing only linearly in the spread of scaling factors and the size of the dataset.","feed_headline":"Axis scaling distorts persistence diagrams by a bounded amount","feed_subtitle":"The claimed limit is half the scaling spread times the dataset diameter.","key_machinery":"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.","core_discovery":"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\n$$\\frac{1}{2}(s_{\\rm avg}-s_{\\min})\\operatorname{diam}(X)\\le d_B(D,D_S)\\le\\frac{1}{2}(s_{\\max}-s_{\\min})\\operatorname{diam}(X),$$\nwhere $s_{\\rm avg}=\\sqrt{\\frac{1}{n}\\sum_i s_i^2}$. 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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the stability theorem for persistence diagrams that the proof invokes to turn a filtration perturbation into a bottleneck-distance bound.","marker":"[3]"},{"why":"Provides the survey-level definitions of persistence diagrams and filtrations that frame the entire argument.","marker":"[2]"},{"why":"Gives the computational construction of persistent homology from filtrations used throughout the paper.","marker":"[1]"},{"why":"Provides an algebraic stability formulation for matching barcodes that underlies the bottleneck-distance comparisons.","marker":"[7]"}],"fun_headline_variants":["Non-uniform scaling warps persistence diagrams by a known bound","Persistence diagrams stay close under axis stretches: explicit bound","How much can coordinate scaling shift persistence diagrams? Bounded!","Bottleneck distance controlled: scaling spread times diameter over two","Stable persistence: non-uniform scaling effects quantified"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Non-uniform scaling warps persistence diagrams by a known bound","Persistence diagrams stay close under axis stretches: explicit bound","How much can coordinate scaling shift persistence diagrams? Bounded!","Bottleneck distance controlled: scaling spread times diameter over two","Stable persistence: non-uniform scaling effects quantified"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001853,"raw_usage":{"total_tokens":7293,"prompt_tokens":977,"completion_tokens":6316,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":6235}},"tokens_in":593,"tokens_out":6316,"duration_ms":42140,"temperature":1.0,"reasoning_tokens":6235,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:33:25.438732+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Persistent homology—a survey ,","cited_arxiv_id":null,"evidence_quote":"Provides the survey-level definitions of persistence diagrams and filtrations that frame the entire argument."},{"cited_title":"Computing persistent homology ,","cited_arxiv_id":null,"evidence_quote":"Gives the computational construction of persistent homology from filtrations used throughout the paper."},{"cited_title":"Induced matchings of barcodes and t he algebraic stability of persistence,","cited_arxiv_id":null,"evidence_quote":"Provides an algebraic stability formulation for matching barcodes that underlies the bottleneck-distance comparisons."}],"review_version":1}