{"id":"c6509681-08a7-4257-bec2-65b6b59c070b","arxiv_id":"2412.03806","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":7,"one_line_summary":"The paper combines McCann interpolation and JKO Wasserstein gradient flow with differentiable persistent homology to iteratively retarget persistence diagrams and update filtrations, but it provides only qualitative 2D examples and no proof that the data follows the claimed Wasserstein dynamics.","lead":"This paper proposes two algorithms that move persistence diagrams toward user-chosen targets by repeatedly re-fitting the data with gradient steps, using McCann interpolation or JKO gradient flow to define each intermediate target. It is a methods preprint with toy examples only: no code, no quantitative evaluation, and the theoretical guarantee behind the Wasserstein dynamics is not established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1's geodesic guarantee is not established: Eq. 13 is a barycentric projection, not a McCann interpolation, and the uniform n-point measure Y(k) generally does not lie on the W2 geodesic from X(k) to Z when n≠m or when the Sinkhorn plan splits mass.","rationale":"The reader's weakest_assumption correctly identifies the load-bearing premise: persistence diagrams are represented as finite, equally weighted empirical measures in R^2, and the geodesic claim rests on treating the Sinkhorn plan and Eq. 13 as a valid discrete McCann interpolation. My analysis agrees with this and makes it more precise: Eq. 13 defines a barycentric projection, and the uniform n-point measure on its output is generally not on the Wasserstein geodesic from the current measure to the target measure. This is not a matter of external consensus but of internal mathematical consistency: the algorithm's own update rule does not implement the McCann interpolation it invokes. A secondary internal inconsistency also supports the rejection: in Algorithm 2, the JKO solution µ obtained in line 7 is never assigned to the y_i variables, so Y(k) in line 8 remains the initial copy of X(k) and the JKO step has no effect on the output. The qualitative experiments, lack of baselines, and absence of code are additional but secondary problems. Since the central claim is unsupported and the algorithms as written contain a concrete mathematical gap, the reader's REJECT verdict stands without modification.","tokens_in":10331,"tokens_out":4606,"duration_ms":52550,"concrete_test":"Run a two-point-to-three-point counterexample: let µ0 = (δ_{(0,0)}+δ_{(1,0)})/2 and ν = (δ_{(-1,1)}+δ_{(1,1)}+δ_{(3,1)})/3. Compute the Sinkhorn plan π, X1 by Eq. 13, and Y at t=1/2. Then compute the W2 distance between the uniform measure on Y and the true McCann geodesic measure µ_{1/2} obtained by solving the OT problem between µ0 and ν and displacing mass along straight segments. If that distance is positive, which it will generally be whenever π has split rows, the claimed geodesic guarantee fails. Repeat with n=m but an entropically regularized Sinkhorn plan that is not a permutation; the same mismatch should appear.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 1 claims to ensure that µt evolves along the W2 geodesic to the target persistence diagram. For this to hold, at each step the update Y(k) = (1−t)X(k)+tX1 must be the McCann interpolation between the empirical measure of X(k) and the target measure Z. But X1 is computed by Eq. 13, x_i = Σ_j π_ij z_j / Σ_j π_ij, which is the barycentric projection of Z through the Sinkhorn plan, not the image of Z under an optimal transport map. Unless π is a permutation matrix with n=m and no split mass, the map T(x_i)=X1_i is not the optimal transport map from µ_k to Z, and uniform(X1) is not Z. McCann's theorem requires a transport map between two measures of equal total mass; when n≠m, the true geodesic measures are not representable as n equally weighted atoms, so the algorithm's restriction to n-point diagrams cannot follow the geodesic. The phrase 'provided that the target persistence diagram is achievable at each step' does not repair this: even when Z is achievable by some filtration, X1 is a projection of Z rather than Z itself, and the actual interpolation is between µ_k and uniform(X1), not toward Z. Thus the central claimed guarantee is not proven and is generally false for split optimal plans.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes two methods for adapting a filtration function on a simplicial complex so that the resulting persistence diagrams evolve in a prescribed way. Algorithm 1 uses Sinkhorn barycenters to define an interpolation between the current persistence diagram and a user-specified target diagram, and then fits the filtration to the interpolated diagram using the critical-set method of Nigmetov and Morozov. Algorithm 2 replaces the target diagram by one obtained from a JKO step of a manually chosen energy functional. Two case studies (circle denoising and circle emerging) are presented, with figures showing point clouds and persistence diagrams over iterations.","tokens_in":1873,"tokens_out":2004,"duration_ms":67826,"significance":"If the central claim were correct, the paper would offer a principled way to invert the map from data to persistence diagrams, enabling direct manipulation of data guided by Wasserstein dynamics in diagram space. The paper builds on recent differentiable-persistence tools (Nigmetov and Morozov; Carriere et al.) and provides illustrative computational experiments. However, as detailed below, the main theoretical guarantee is not proven and is in fact false for the algorithm as written, and the second algorithm's pseudocode does not implement the claimed JKO dynamics. The paper also lacks any quantitative evaluation. The strengths are the clear presentation of background material and the use of state-of-the-art computational components (Sinkhorn, sliced Wasserstein, big-step topological optimization), but these do not compensate for the load-bearing rigor gaps.","major_comments":[{"comment":"The claimed guarantee that μ_t evolves along the W2-geodesic to Z is not established and is generally false. The update X1_i = (Σ_j π_ij z_j)/(Σ_j π_ij) is the barycentric projection of the target through the Sinkhorn plan, not the image of Z under the optimal transport map. McCann's theorem applies to the interpolation between μ_k and the push-forward of μ_k under the optimal map; it does not apply to a barycentric projection unless π is a permutation matrix with n=m and no split mass. In the typical case n≠m, X1 is not a measure equal to Z and the interpolation is between μ_k and uniform(X1), not toward Z. Moreover, even when Z is achievable, Equation (13) projects each source point to a weighted average of target points; the resulting empirical measure is not Z, so the sequence X(k) does not converge to Z in general. The phrase 'provided that the target persistence diagram is achievable at each step' is also circular because the paper gives no characterization of achievability and the authors explicitly state (Section 4.1) that the learnable scheme cannot guarantee that the persistence diagram is always achievable. Thus the geodesic guarantee is not proven and cannot be checked.","section":"Section 4.1, Eq. (13), Algorithm 1 lines 5-7"},{"comment":"The pseudocode does not propagate the JKO solution to Y(k). Line 4 initializes (y_1,...,y_n) to (x_1,...,x_n); line 7 computes a new measure μ as the argmin; but line 8 then sets Y(k) = {y_i^(k)}, the unchanged initial values. As written, Y(k) = X(k) at every iteration, so the inner filtration-fitting loop fits the current diagram to itself and no dynamics occur. If the intended assignment was to take the support of the argmin measure, this must be stated explicitly, together with a proof that this support forms a valid persistence diagram and that the resulting cardinality can change. Without such a correction, Algorithm 2 is not a meaningful implementation of the JKO scheme.","section":"Section 4.2, Algorithm 2 lines 4-8"},{"comment":"The fixed-cardinality formulation is incompatible with the circle-emerging experiment. Algorithm 2 initializes the target measure with the same n points as the current diagram and, if restricted to n-point measures, the JKO step can only move existing points; it cannot create new off-diagonal persistence pairs. Yet the circle-emerging experiment (Section 5.2) aims to create H1 features that are absent from the initial random point cloud. The paper never explains how the extra H1 pairs appear, so the reported results cannot be attributed to the algorithm as described. This is a fundamental mismatch between the algorithm's representational constraints and the experimental goal.","section":"Section 4.2 and Section 5.2"},{"comment":"The paper models a persistence diagram as an empirical measure with total mass 1 on the off-diagonal points. This is not the standard Wasserstein geometry of persistence diagrams, where the diagonal is a separate set with its own multiplicity and mass can be transported to and from the diagonal. Consequently, the 'Wasserstein gradient flow' discussed is a flow in P2(R^2), not a flow in the space of persistence diagrams under the usual W2 metric. The authors neither justify this choice nor relate it to stability properties of persistence diagrams, so the topological relevance of the construction is left unjustified.","section":"Sections 3.3 and 4.1"},{"comment":"There are no quantitative results. The case studies are qualitative figure sequences; no convergence curves, error bars, runtime measurements, or comparisons with prior methods are provided. The hyperparameters t_k, η, τ, ϵ, and the energy functionals are not listed, and no sensitivity analysis is given. Thus the empirical claims cannot be verified or reproduced from the manuscript.","section":"Section 5"}],"minor_comments":[{"comment":"There is a typo: 'analyais' should be 'analysis'.","section":"Section 1"},{"comment":"The sentence 'Recent work has shifted focus to the use of p-Wasserstein distances, A central challenge...' has a capitalization and run-on issue; it should be split into two sentences.","section":"Section 2"},{"comment":"The JKO formula displays 'W 2 2 (µ, µk)' with inconsistent sub/superscript formatting; it should be written as W_2^2(µ, µk) throughout.","section":"Section 3.4"},{"comment":"The repulsion loss formula is ambiguous: the expression '1/∥σi − σj∥2 + ϵ' lacks parentheses; the intended formula is likely Σ_i Σ_{j≠i} 1/(∥σi − σj∥^2 + ϵ).","section":"Section 5.1.1"},{"comment":"The symbol f is used both for the filtration function and for the update 'f ← f − η∇f'; the update is not clearly defined for the simplicial complex K, since the critical-set method of Nigmetov and Morozov returns a filtration function, but the manuscript does not specify how the update is applied to all simplices.","section":"Algorithms 1 and 2"},{"comment":"The target for the 1st persistence diagram in the circle-emerging experiment is not specified explicitly; the text says 'enhance the circles' but gives no birth-death coordinates, unlike the 0th diagram target (0, 0.08). A precise target would improve reproducibility.","section":"Section 5.2"},{"comment":"The reference 'Wang et al.' in the Related Work section is incomplete: it lacks a year, venue, and full author list.","section":"References"}],"recommendation":"reject","confidential_remarks":"The paper's central algorithmic claims are not supported by the mathematics as written. The geodesic guarantee in Algorithm 1 is false for the stated computation, and Algorithm 2's pseudocode does not execute a JKO step at all. The fixed-cardinality issue further contradicts the circle-emerging experiment. These are not merely presentation issues; they concern the core contribution. I see no straightforward local fix that would keep the manuscript's scope and central claim intact, so I recommend rejection rather than major revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the idea is plausible but the paper does not support it. What is actually new is coupling OT-derived intermediate persistence diagrams (McCann interpolation or JKO steps) with the Nigmetov-Morozov big-step critical-set optimizer to back-propagate diagram changes into filtration values. That is a real extension of Gameiro et al. and Carriere et al., and the two toy experiments show the intended behavior. I also give credit for a limitations section that names missing statistical theory, GPU scalability, and unresolved singleton-loss conflicts.\n\nThe soft spots are load-bearing. Algorithm 1 claims the measure µt moves along the W2 geodesic to the target diagram. But the update uses the barycentric projection of the Sinkhorn plan, Eq. 13, not the McCann interpolation. McCann needs an optimal transport map between two measures of equal mass; when n≠m or the Sinkhorn plan splits mass, X1 is not the target and uniform(X1) is not Z. So the algorithm is interpolating toward a projection of the target, not the target, and the claimed geodesic property is unproved and generally false. The caveat \"provided the target is achievable\" does not fix this.\n\nAlgorithm 2 has a more basic problem. The number of atoms is fixed at n for all k, so the measure update cannot create or destroy off-diagonal persistence pairs. The circle-emerging experiment depends on changing which H1 features survive, so the mechanism cannot be doing what the figure suggests. Worse, the pseudocode initializes y_i from x_i and never updates y_i after the JKO step; line 8 sets Y(k) to the stale y_i, so Algorithm 2 as written returns Y(k)=X(k). That is an internal contradiction, not a minor typo, because the JKO solution on line 7 is discarded.\n\nThe experiments are qualitative: two 2D examples, no baselines, no error bars, no code or data. For a methods paper that's thin even when the theory is solid. Here the theory is not solid.\n\nOverall: I see a worthwhile heuristic and a lot of work needed. The framing of persistence diagrams as equally weighted empirical measures in R^2 is a mismatch for the persistence-diagram metric with its diagonal, and the paper never confronts that. The idea might be salvageable by restricting to equal-cardinality matched diagrams or by explicitly treating the diagonal, and then proving when the critical-set optimizer can realize the interpolation. As it stands, I would not send this to peer review; I'd desk reject and suggest the authors resubmit after fixing the algorithms and adding quantitative evidence.","headline":"Pleasantly motivated but not ready: the geodesic guarantee is unproved, Algorithm 2's pseudocode returns a constant sequence, and the experiments are qualitative.","tokens_in":11195,"tokens_out":3481,"would_cite":false,"duration_ms":37802,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper claims that persistence diagrams can be driven along Wasserstein geodesics by editing the underlying data through a gradient update of the filtration.","keywords":["persistent homology","persistence diagram","Wasserstein gradient flow","optimal transport","McCann interpolation","JKO scheme","topological data analysis","filtration optimization"],"falsifier":"Count the number of off-diagonal birth–death pairs in the 1st persistence diagram while Algorithm 2 runs on uniformly random points; because the JKO target is initialized with the same number of atoms as the current diagram, the count cannot increase, so a genuinely new circle pair emerging from the diagonal would contradict the algorithm's stated mechanism.","tokens_in":10045,"feed_emoji":"🌀","tokens_out":4744,"duration_ms":44913,"temperature":0.7,"pith_summary":"The paper tries to establish that variations in persistence diagrams can be translated back into changes of the underlying data, by treating the diagram as a probability measure and letting it evolve along a Wasserstein gradient flow. If true, this replaces the one-way data-to-diagram pipeline with a loop in which a user prescribes a target diagram (or an energy) and the data is edited so its topological summary follows a geodesic in Wasserstein space. The two proposed algorithms—one based on McCann interpolation toward a fixed target, one based on the JKO scheme with an energy functional—each couple the diagram's motion to the filtration by a differentiable critical-set update. The paper demonstrates the idea on circle denoising and circle emergence, and identifies scalability, statistical theory, and loss conflicts as open limitations.","feed_headline":"Tune data by flowing its persistence diagram along W2 geodesics","feed_subtitle":"Two algorithms translate persistence-diagram changes back into edits of the original point cloud, graph, or image.","key_machinery":"The central objects are McCann interpolation—the displacement interpolation between two probability measures along the optimal transport map, which yields a geodesic in Wasserstein space—and the Jordan–Kinderlehrer–Otto (JKO) scheme, a time-discretized Wasserstein gradient flow. Persistence diagrams are represented as equally weighted empirical measures in the Euclidean plane, and the critical-set gradient method supplies the update that turns an abstract diagram movement into a concrete change of the filtration function on the simplicial complex.","core_discovery":"The central claim is that the one-way pipeline from data to persistence diagram can be reversed by viewing a persistence diagram as an empirical probability measure with equal weights at the birth–death points and running a Wasserstein gradient flow on it. Given a target diagram, the first algorithm uses the optimal transport plan between the current diagram and the target to form a McCann displacement interpolation, then adjusts the filtration function with a critical-set gradient method so the data's computed diagram tracks the interpolated diagram. Without a target, the second algorithm applies one JKO step of an energy functional to the current diagram, then performs the same filtration update. The paper demonstrates both schemes on a noisy circle (denoising) and on random points (circle emergence), claiming the diagrams evolve along $W_2$ geodesics whenever the targets are achievable at each step.","pith_inferences":["An extension the paper leaves implicit: because the JKO algorithm fixes the number of persistence pairs, any apparent creation or destruction of off-diagonal pairs must actually come from the energy functional moving near-diagonal pairs across the diagonal; a direct test would be to count off-diagonal pairs over time.","The same flow formulation could be applied to other persistence summaries, such as zigzag or multiparameter persistence, if a suitable metric and a differentiable projection back to data are available.","Replacing the exact optimal transport plan with an entropy-regularized Sinkhorn plan would give a smoothed McCann interpolation, potentially improving numerical stability for large diagrams.","The guarantee degrades gracefully if the target diagram is not achievable at every step; measuring the gap between the interpolated diagram and the diagram actually realized by the updated filtration would quantify how much of the geodesic property survives."],"forward_implications":["With a known target diagram, the McCann-based algorithm makes the persistence diagram evolve along a Wasserstein geodesic toward the target whenever the target is reachable at every step.","Without a target diagram, the JKO-based algorithm lets an energy functional steer the persistence diagram, initializing the target at the current diagram so the number of persistence pairs is fixed.","The filtration update via the critical-set method means each optimization step edits the original data (point cloud, graph, or image) rather than the diagram directly.","In the circle-denoising case, combining the two algorithms removes noise from the 0th persistence diagram and fills a gap in the 1st persistence diagram, making the underlying circle explicit.","In the circle-emerging case, uniformly random points are rearranged so a first-homology circle appears, showing the method can amplify a weak topological signal."],"supporting_citations":[{"why":"Provides the Wasserstein gradient flow framework and the Γ-convergence result that links the JKO scheme to the continuous flow.","marker":"(Ambrosio et al., 2008)"},{"why":"Supplies the dynamic formulation of optimal transport used as Theorem 1 in the paper.","marker":"(Benamou and Brenier, 2000)"},{"why":"Introduces the JKO scheme, the discrete minimizing-movement approximation of Wasserstein gradient flow used in Algorithm 2.","marker":"(Jordan et al., 1998)"},{"why":"Provides the Sinkhorn algorithm used in Algorithm 1 to compute the optimal transport plan efficiently.","marker":"(Cuturi, 2013)"},{"why":"Supplies the critical-set gradient method that differentiates persistence-diagram losses and updates the filtration function in both algorithms.","marker":"(Nigmetov and Morozov, 2024)"},{"why":"Is the source for the McCann interpolation theorem quoted as Theorem 2, the core geometric object of Algorithm 1.","marker":"(Villani, 2021)"}],"fun_headline_variants":["Reverse-engineer persistence diagrams to edit point clouds","Flow persistence diagrams to reshape data directly","Wasserstein flow turns diagram changes into data edits","Edit data by pushing its persistence diagram along W2","From diagram to data: Wasserstein gradient flow algorithms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework assumes persistence diagrams are just finite equally weighted point sets in the plane, that the standard $W_2$ metric governs their motion, and that the desired target diagram can actually be reached at each optimization step.","fun_headline_variants_meta":{"raw":{"variants":["Reverse-engineer persistence diagrams to edit point clouds","Flow persistence diagrams to reshape data directly","Wasserstein flow turns diagram changes into data edits","Edit data by pushing its persistence diagram along W2","From diagram to data: Wasserstein gradient flow algorithms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000347,"raw_usage":{"total_tokens":1807,"prompt_tokens":758,"completion_tokens":1049,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":374,"completion_tokens_details":{"reasoning_tokens":987}},"tokens_in":374,"tokens_out":1049,"duration_ms":7305,"temperature":1.0,"reasoning_tokens":987,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:05:02.605415+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Count the number of off-diagonal birth–death pairs in the 1st persistence diagram while Algorithm 2 runs on uniformly random points; because the JKO target is initialized with the same number of atoms as the current diagram, the count cannot increase, so a genuinely new circle pair emerging from the diagonal would contradict the algorithm's stated mechanism.","supporting_citations":[],"review_version":1}