{"id":"ff6d4985-76dc-4d48-9537-a71d7e5bde50","arxiv_id":"2507.14340","paper_version":1,"verdict":"REJECT","confidence":"LOW","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The Polar Persistence Distance (PPD) is a quasi-metric for persistence diagrams that adds angular separation to radial distance, and the paper applies it to social choice preference data.","lead":"This paper defines a polar-coordinate distance for persistence diagrams, called the Polar Persistence Distance (PPD), pairing radial distance with angular separation. It proposes applying persistent homology to social choice data, such as election rankings, and gives experiments claiming PPD is more discriminative than bottleneck or Wasserstein distances.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"PPD has no stable diagonal: near-origin points retain positive distance to the diagonal, so diagrams converging to empty do not converge in PPD, contradicting the core noise-robustness claim.","rationale":"Reading in good faith, the pointwise PPD is indeed symmetric, non-negative, and smooth away from the origin, and the paper correctly avoids claiming the triangle inequality by calling it a quasi-metric. The scaling property in Proposition 4.9 is also correct. However, the paper's headline contribution is a 'noise-robust polar distance' for persistence diagrams, and that requires the distance to be continuous with respect to the usual convergence of diagrams, including matching points to the diagonal. The provided stability theorem (Theorem 4.5) assumes a fixed bijection and a bounded region, which excludes exactly the diagonal-matching regime where persistence diagrams are compared in practice. The concrete near-origin example shows the failure: points with small coordinates but fixed angle have a positive PPD to the diagonal, so a sequence of diagrams converging to the empty diagram under the standard bottleneck or Wasserstein metric does not converge under PPD. This directly refutes the central robustness claim, independent of the weak empirical evidence. The reader's weakest assumption identified the angular instability near the origin and the insufficiency of Theorem 4.5, so there is partial agreement; the reader framed it as an untested assumption, whereas the analysis here shows it is a concrete discontinuity. The experiments in Section 5 report only two scalar distances without error bars, statistical tests, or a defined diagram-level PPD with diagonal matching, so they do not provide independent support for robustness. A verdict of REJECT is therefore appropriate, and the reader's REJECT verdict should stand unchanged.","tokens_in":19173,"tokens_out":6543,"duration_ms":80930,"concrete_test":"Compute d_PPD between D_t={(t,2t)} and the empty diagram (or between {(t,2t)} and {(2t,3t)}) for t=1, 0.1, 0.01, 0.001 with alpha=1, using the paper's formula in Definition 4.1. If the distance does not approach 0 as t->0, then PPD is discontinuous at the diagonal and the noise-robustness claim fails. A minimal Python implementation suffices; no matching is needed for this single-point case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central advertised property is noise robustness, but the Polar Persistence Distance is discontinuous on the standard space of persistence diagrams because it has no stabilizing distance to the diagonal. Theorem 4.5 assumes a fixed bijection between diagrams with the same number of off-diagonal points, all in a bounded region avoiding the origin; it does not cover the matching-to-diagonal paradigm used by every standard diagram distance, nor does it handle points approaching the origin. This is not merely a technical gap. For p_t=(t,2t), r_p=sqrt(5)t and theta_p=atan2(2,1) about 63.4 degrees. The infimum of PPD^2(p_t,(s,s)) over diagonal points (s,s) is attained at s=sqrt(5/2)t and equals alpha*sin^2((theta_p-pi/4)/2)>0. Hence d_PPD({p_t}, empty diagram) does not tend to 0 as t->0, even though p_t converges to the diagonal. Similarly, d_PPD({(t,2t)},{(2t,3t)}) tends to a positive constant as t->0 because the angular difference is about 7.1 degrees, so two diagrams that both converge to the empty diagram remain O(1) apart. The masking remark in Section 4.9 introduces a discontinuous cutoff rather than resolving the issue. Consequently, the abstract's claim that PPD addresses 'instability under perturbation' is unsupported and, in the standard diagram-metric topology, false. The two scalar-valued experiments in Sections 5.4 and 5.6 cannot compensate, since they compare only single pairs of diagrams without specifying a diagram-level matching protocol or providing code.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a new pointwise distance on persistence-diagram points, called the Polar Persistence Distance (PPD), defined in Definition 4.1 as sqrt((r1-r2)^2 + alpha * sin^2((theta1-theta2)/2)), where (r,theta) are polar coordinates of a birth-death point. The authors claim that this quasi-metric is smooth, differentiable, and noise-robust, that it captures angular structure missed by bottleneck and Wasserstein distances, and that it enables a new field of 'Topological Social Choice'. The paper reviews several existing persistence-diagram distances, proves local smoothness and a restricted stability result, and reports pairwise distance values between two subsets of the Irish election and sushi preference datasets, interpreting the larger PPD values as evidence of greater discriminative power.","tokens_in":19446,"tokens_out":6288,"duration_ms":75408,"significance":"The motivating problem is real: differentiable diagram distances are useful for gradient-based learning, and angular information may be informative in some applications. The paper also states a clear mathematical formula and identifies a genuine gap in the existing menu of ML-friendly diagram distances. However, the central claims are not established. The PPD is never defined as a distance on the standard space of persistence diagrams with diagonal matching; it is discontinuous in the usual diagram topology; the stability theorem assumes a fixed bijection with equal cardinality and a bounded region avoiding the origin; and the empirical section consists of three scalar values per dataset with no error bars, no statistical tests, and no task-based baseline. The claims of noise robustness and empirical superiority are therefore unsupported, and the internal contradiction with the paper's own statements of classical stability theorems further weakens the motivation.","major_comments":[{"comment":"Definition 4.1 defines PPD only for pairs of off-diagonal points, and the only diagram-level statement, Theorem 4.5, assumes two diagrams with the same number of off-diagonal points and a fixed bijection between them. No matching-to-diagonal or variable-cardinality extension is formally defined. The paper mentions in §4.5 that 'diagram-level extensions of PPD using optimal matching' are possible, but no such definition is given, and Section 5.3 does not specify the matching protocol used in the experiments. Since persistence diagrams in the standard space have different cardinalities and include points arbitrarily close to the diagonal, d_PPD is not actually defined on the objects compared in Tables 2 and 3. This is a load-bearing gap: the central object of the paper is a pointwise formula, not a distance on the space of persistence diagrams.","section":"§4.2, §4.4, §5.3"},{"comment":"The claimed noise robustness is contradicted by the behavior near the origin. For p_t=(t,2t), the infimum of PPD^2(p_t,(s,s)) over diagonal points (s,s) is attained at s=sqrt(5/2)t and equals alpha*sin^2((atan2(2,1)-pi/4)/2)>0. Hence d_PPD({p_t}, empty diagram) does not tend to 0 as t->0, even though {p_t} converges to the empty diagram in the bottleneck and Wasserstein metrics. Similarly, d_PPD({(t,2t)},{(2t,3t)}) tends to a positive constant as t->0. The masking remark in Section 4.9 introduces a small exclusion radius epsilon, which is a discontinuous cutoff rather than a stabilizing term. Consequently, the abstract's claim that PPD addresses 'instability under perturbation' is not merely unproven; in the standard diagram-metric topology it is false. The stability theorem 4.5 is also stated without proof, and its assumptions exclude the near-origin and diagonal-matching cases that are essential for noise robustness.","section":"§4.4, §4.9"},{"comment":"The empirical support is anecdotal. Table 2 reports three scalar values for one pair of subsets of the Irish dataset, and Table 3 reports three scalar values for one pair of subsamples of the sushi dataset. There are no error bars, no repeated trials, no statistical tests, and no comparison against a downstream task such as classification, retrieval, or clustering. The interpretation in Section 6 that a higher PPD value implies 'greater discriminative power' is an assertion, not a demonstrated property. Without a ground-truth label or a task-based evaluation, larger distance values could simply reflect the metric's scale, the value of alpha, or the undefined matching protocol. These experiments cannot support the paper's empirical claims of superiority.","section":"§5.4, §5.6, §6"},{"comment":"The motivation states that bottleneck and Wasserstein distances exhibit 'instability under perturbation' and are 'sensitive to noise,' yet Theorems 3.2 and 3.4 in the same paper state their stability under bounded perturbations of tame functions. This internal contradiction is load-bearing because noise robustness is presented as the primary advantage of PPD. The paper should either compare against the known stability guarantees or reframe the claimed advantage in terms of differentiability and angular sensitivity, rather than asserting a property that the paper itself proves classical distances possess.","section":"§3.2–§3.3, Abstract, §1"},{"comment":"Lemma 6.1 is misleading. For p1=(r,0) and p2=(r cos theta, r sin theta), the Euclidean distance is 2r|sin(theta/2)|, which increases with theta and does not 'collapse to zero' for angular misalignment. The lemma's own computation shows that classical Euclidean-coordinate distances do detect the rotation through the point displacement. Thus the claim that bottleneck and Wasserstein distances are insensitive to angular structure is not established by this example. The example conflates rotation-invariance of the metric with insensitivity to rotated configurations.","section":"§6, Lemma 6.1"}],"minor_comments":[{"comment":"The list of contributions contains '(iv) Empirical Relevance' twice, and the introduction states that Section 3 introduces the PPD, but Definition 4.1 appears in Section 4.","section":"§1"},{"comment":"The section title should be 'Persistent Homology,' and the formula 'H_k(Kϵ' in the filtration paragraph is missing a closing parenthesis.","section":"§2.1"},{"comment":"The Sliced Wasserstein distance is defined twice, in Definitions 3.3 and 3.6, with inconsistent notation: the projection is written as x cos theta + y sin theta in one place and as the inner product <p,theta> in another, and the integration domain is [0,pi] versus S^1.","section":"§3.4, §3.6"},{"comment":"Table 2 does not state the value of alpha used, while Table 3 uses alpha=1.5; since alpha is a free parameter controlling the angular term, the reported numbers are not reproducible without this information.","section":"§5.3, §5.4"},{"comment":"The claim that the Polar Gaussian kernel 'satisfies Mercer's condition in common cases' is said to be supported by 'preliminary experiments,' but no experiments or conditions are provided; this statement should be removed or substantiated.","section":"§4.8"}],"recommendation":"reject","confidential_remarks":"The manuscript has the character of an early preprint rather than a finished research paper. The mathematical core is a simple formula whose diagram-level definition is missing, and the experimental section does not meet the standard of evidence for the claimed empirical superiority. The novelty claim of being the first to apply persistent homology to social choice is likely overstated, and the paper does not engage with existing work on differentiable diagram distances beyond a list of kernels. If the authors are encouraged to resubmit, they would need to define a genuine diagram-level distance with diagonal matching, prove or disprove its continuity in the standard diagram topology, and replace the current two-table comparison with a full experimental study."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The one thing you should know before reading: the paper's main advertised property, noise robustness, is false as stated. For a point p_t=(t,2t), the infimum of PPD^2 to diagonal points (s,s) is alpha*sin^2((theta_p - pi/4)/2), which is positive and independent of t. So d_PPD({p_t}, empty) does not go to 0 as t goes to 0, breaking the standard persistence-diagram topology. The masking remark in Section 4.9 is a patch, not a fix.\n\nWhat is actually new: the distance formula itself—a polar-coordinate quasi-metric with a sin^2 half-angle angular term—does not appear in the cited literature. The paper also correctly notes that bottleneck and Wasserstein ignore angular orientation, and the Section 3 survey of diagram metrics is a competent overview. The social-choice framing is new but shallow.\n\nSoft spots, in order: (1) the diagonal instability above is load-bearing; Theorem 4.5 assumes a fixed bijection and a bounded region avoiding the origin, which sidesteps the problem and does not give the standard matching-to-diagonal stability. (2) The experiments are two tables of single scalar distances, no error bars, no repetitions, no statistical test, no matching protocol, and no baseline task. The claim that a larger distance means greater discriminative power is untested and not self-evident. (3) The paper overclaims \"extensive experiments\" and \"superiority.\" There are also copy-paste errors (duplicated contribution item, repeated words) that suggest the manuscript is not polished.\n\nThe mathematical definition is clear and the authors engage with relevant literature, but the central robustness property is violated. The metric as defined is not a distance on the space of persistence diagrams under the usual diagonal-matching paradigm.\n\nWho gets value? Someone curious about alternative diagram distances might read this for the formula, but they would need to fix the diagonal problem and run real experiments. As it stands, it is not a usable contribution.\n\nRecommendation: desk reject. The gap between claims and evidence is large, and the diagonal issue undercuts the paper's stated purpose. If the authors redefine PPD so that diagonal points act as a proper sink (e.g., by setting the angular term to zero on the diagonal) and provide rigorous experiments, it could be reconsidered.","headline":"The PPD formula is genuinely new, but the paper's central noise-robustness claim fails because the distance has no stable diagonal: points near the origin keep a positive distance to the diagonal, so diagrams converging to empty do not converge.","tokens_in":20013,"tokens_out":2977,"would_cite":false,"duration_ms":36238,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["55N31","62R40","91B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that comparing persistence diagrams in polar coordinates—radius for size, angle for direction—through a smooth, tunable quasi-metric separates voting and preference profiles that classical bottleneck and Wasserstein…","keywords":["topological data analysis","persistence diagrams","persistent homology","polar coordinates","quasi-metric","social choice theory","preference aggregation","differentiable distance"],"falsifier":"Take two pairs of preference datasets with known labels where the persistence diagrams have identical radial coordinates but rotated angular coordinates, and compare classification accuracy using PPD-based kernels versus Wasserstein- or bottleneck-based kernels on held-out data; if the angular term adds no predictive signal beyond radial and Euclidean comparisons, the central discrimination claim fails.","tokens_in":18891,"feed_emoji":"🗳️","tokens_out":11258,"duration_ms":602119,"temperature":0.7,"pith_summary":"The paper proposes the Polar Persistence Distance (PPD), which compares persistence-diagram points by polar coordinates—radius $r=\\sqrt{b^2+d^2}$ and angle $\\theta=\\operatorname{atan2}(d,b)$—rather than by their Euclidean coordinates. Its formula is $\\sqrt{(r_1-r_2)^2+\\alpha\\sin^2((\\theta_1-\\theta_2)/2)}$, with $\\alpha>0$ tuning how strongly angular differences count. The authors argue that classical bottleneck and Wasserstein distances ignore this angular structure, are not differentiable, and are poorly suited to learning pipelines, whereas PPD is smooth, locally Lipschitz, and a quasi-metric (no triangle inequality). On Irish election and sushi preference data, PPD returns larger pairwise distances between persistence diagrams of different voter sub-profiles, which the authors read as finer discrimination of dominance cycles and consensus clusters; if this holds, persistence diagrams become usable, differentiable inputs for predicting collective decisions, a direction the paper names 'topological social choice'.","feed_headline":"Polar distance adds angular sensitivity to persistence comparisons","feed_subtitle":"The new distance weighs angular displacement, not just radius, when comparing preference diagrams.","key_machinery":"The carrying object is the formula $d_{\\mathrm{polar}}(p_1,p_2)^2=(r_1-r_2)^2+\\alpha\\sin^2((\\theta_1-\\theta_2)/2)$, where $r_i=\\sqrt{b_i^2+d_i^2}$ and $\\theta_i=\\operatorname{atan2}(d_i,b_i)$ for persistence points $p_i=(b_i,d_i)$, and $\\alpha>0$ is a tunable weight. The radial term $|r_1-r_2|$ measures difference in overall birth–death magnitude; the half-angle sine term measures angular separation on the unit circle in a smooth, $2\\pi$-periodic way. This angular term is what distinguishes PPD from Euclidean metrics, and the differentiability of both terms is what makes PPD compatible with gradient-based optimization. The stability, scaling, and local-Lipschitz results in Section 4 all follow from the smoothness of these two components.","core_discovery":"The central claim is that the Polar Persistence Distance (PPD), $d_{\\mathrm{polar}}(p_1,p_2)=\\sqrt{(r_1-r_2)^2+\\alpha\\sin^2((\\theta_1-\\theta_2)/2)}$, is a quasi-metric—symmetric, non-negative, zero only for equal points, but without the triangle inequality—on off-diagonal persistence points, and that it captures both the magnitude and the orientation of topological features in a way that makes persistence-diagram comparisons more discriminative for social choice data than bottleneck or Wasserstein distances. The paper proves that the polar-coordinate map is smooth away from the origin, that PPD is continuously differentiable in its inputs, and that it is locally Lipschitz; it also proves stability under small perturbations only when the diagrams have the same number of off-diagonal points and are matched by a fixed bijection inside a bounded region (Theorem 4.5). The paper does not claim the triangle inequality, and it explicitly identifies PPD as a quasi-metric. Empirically, on the Irish election and sushi preference datasets, PPD yields larger pairwise distances between sub-profile diagrams than either Wasserstein or bottleneck distances, and the paper interprets this as evidence that angular displacement captures rotations or realignments of dominance cycles that Euclidean metrics collapse. That interpretation—larger values mean greater sensitivity and expressiveness, not instability—is the paper's own framing.","pith_inferences":["Editorial inference: The paper's reading of larger PPD values as 'greater discriminative power' is an interpretation, not a proven property; a direct test would be to check whether PPD-separated diagrams correspond to behaviorally distinct profiles under an external label.","Editorial inference: Since $\\theta=\\operatorname{atan2}(d,b)$ is unstable near the origin and the paper masks near-diagonal points, the claimed robustness should be tested against the choice of masking radius; results that change sharply with that radius would undercut the noise-robustness claim.","Editorial inference: PPD's failure of the triangle inequality may affect clustering and geodesic computations; a natural extension is to measure how often and how severely triangle inequality violations occur in practice and whether they change downstream model rankings.","Editorial inference: Because the empirical comparisons use only two real datasets and single sub-profile splits, an obvious extension is a controlled simulation where preference profiles are rotated or perturbed with known ground truth, to quantify how much of PPD's angular sensitivity is signal rather than noise."],"forward_implications":["If PPD behaves as claimed, persistence-diagram distances can be used directly as differentiable loss terms in neural networks, which bottleneck and Wasserstein distances cannot.","Social choice comparisons gain an angular axis: two profiles whose topological features differ mainly by rotation in the birth–death plane will be separated by PPD even when their Euclidean summaries nearly coincide.","A Gaussian kernel built from PPD, $K_{\\mathrm{polar}}(p_1,p_2)=\\exp(-d_{\\mathrm{polar}}^2/2\\sigma^2)$, gives a route to kernel-based machine learning over diagrams if Mercer's condition holds in the cases claimed.","The empirical pipeline—dominance matrix from rankings, one-skeleton filtration, persistence diagrams, then PPD comparison—offers a template for forecasting collective outcomes from raw preference data.","The paper's proposal of 'topological social choice' as a field implies that voting systems can be compared by the persistent topology of their preference complexes, not only by aggregated outcomes."],"supporting_citations":[{"why":"Supplies the standard definition of persistence diagrams and persistent homology, the objects PPD compares.","marker":"[12]"},{"why":"Provides the classical stability theorem for persistence diagrams that motivates and benchmarks the new distance.","marker":"[7]"},{"why":"Supplies the bottleneck-matching algorithms whose computational cost and nondifferentiability motivate a differentiable alternative.","marker":"[8]"},{"why":"Defines the Wasserstein framework on persistence diagrams used as the classical baseline in experiments.","marker":"[9]"},{"why":"Provides the simplicial-complex and persistence computation library used to build filtrations and diagrams in the experiments.","marker":"[5]"},{"why":"Supplies the real-world ranking datasets (Irish election, sushi) used for the empirical comparisons.","marker":"[29]"},{"why":"Documents cyclic and contextual structure in sushi preferences that the paper cites to interpret PPD's angular sensitivity.","marker":"[28]"},{"why":"Establishes the stability baseline for persistence diagrams invoked when the paper argues classical metrics are noise-sensitive in learning settings.","marker":"[1]"}],"fun_headline_variants":["Polar metric captures angular shifts in preference topology","New distance weighs angle, not just radius, for persistence","Angular-aware persistence distance for noisy social choice","Quasi-metric polar distance for robust TDA in voting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the angular coordinate $\\theta=\\operatorname{atan2}(d,b)$ of a birth–death point carries meaningful, stable structure for comparing preference diagrams; this coordinate is highly sensitive near the origin and under small perturbations, and the paper's stability theorem assumes a fixed bijection over a bounded region rather than general diagram matching.","fun_headline_variants_meta":{"raw":{"variants":["Polar metric captures angular shifts in preference topology","New distance weighs angle, not just radius, for persistence","Angular-aware persistence distance for noisy social choice","Quasi-metric polar distance for robust TDA in voting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000192,"raw_usage":{"total_tokens":1402,"prompt_tokens":1058,"completion_tokens":344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":674,"completion_tokens_details":{"reasoning_tokens":280}},"tokens_in":674,"tokens_out":344,"duration_ms":4668,"temperature":1.0,"reasoning_tokens":280,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:56:45.484484+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take two pairs of preference datasets with known labels where the persistence diagrams have identical radial coordinates but rotated angular coordinates, and compare classification accuracy using PPD-based kernels versus Wasserstein- or bottleneck-based kernels on held-out data; if the angular term adds no predictive signal beyond radial and Euclidean comparisons, the central discrimination claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard definition of persistence diagrams and persistent homology, the objects PPD compares."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the classical stability theorem for persistence diagrams that motivates and benchmarks the new distance."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the bottleneck-matching algorithms whose computational cost and nondifferentiability motivate a differentiable alternative."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Wasserstein framework on persistence diagrams used as the classical baseline in experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the simplicial-complex and persistence computation library used to build filtrations and diagrams in the experiments."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the real-world ranking datasets (Irish election, sushi) used for the empirical comparisons."},{"cited_title":"(2005, June)","cited_arxiv_id":null,"evidence_quote":"Establishes the stability baseline for persistence diagrams invoked when the paper argues classical metrics are noise-sensitive in learning settings."}],"review_version":1}