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REVIEW 5 major objections 5 minor 28 references

Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2507.14340 v1 pith:K5JZ3774 submitted 2025-07-18 math.AT cs.DScs.LG

Athanasios Andrikopoulos , Nikolaos Sampanis This is my paper · ORCID
classification math.ATcs.DScs.LG MSC 55N3162R4091B12
keywords topologicaldataanalysispersistencediagramspersistenthomologypolarcoordinatesquasi-metricsocialchoicetheorypreferenceaggregationdifferentiabledistance
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

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'.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

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.

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 (5)
  1. [§4.2, §4.4, §5.3] 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.
  2. [§4.4, §4.9] 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.
  3. [§5.4, §5.6, §6] 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.
  4. [§3.2–§3.3, Abstract, §1] 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.
  5. [§6, Lemma 6.1] 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.
minor comments (5)
  1. [§1] 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.
  2. [§2.1] The section title should be 'Persistent Homology,' and the formula 'H_k(Kϵ' in the filtration paragraph is missing a closing parenthesis.
  3. [§3.4, §3.6] 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.
  4. [§5.3, §5.4] 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.
  5. [§4.8] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: PPD is an independently defined quasi-metric whose stated smoothness and stability properties are argued directly from the definition, and the only self-citation is incidental background context.

full rationale

The Polar Persistence Distance is defined in Definition 4.1 with a free hyperparameter alpha that is not fitted to the experimental data; the Sushi study simply fixes alpha=1.5, and no fitting or calibration procedure appears in the paper. The differentiability and local Lipschitz results (Lemma 4.2, Theorem 4.3, Proposition 4.4) are derived by direct calculus from the coordinate map and the closed-form expression for PPD, not from any fitted quantity. Theorem 4.5 states a stability bound under an explicit fixed-bijection and bounded-region assumption; this is an independently stated hypothesis rather than a prediction that reduces to a fit. The only self-citation is Sampanis (2025) in the biology background of Section 2.1, which is not load-bearing for any PPD claim. The interpretation of larger PPD values in Tables 2 and 3 as 'richer differentiation' is an empirical claim whose support is weak, since PPD has an extra positive angular term by construction, but that is a correctness and validation concern rather than circularity. Likewise, Section 4.9's near-origin masking caveat and the absence of a stable diagonal for points converging to the origin are mathematical gaps in the advertised noise robustness, not circular reductions. No equation, theorem, or experimental conclusion in the paper is equivalent by construction to its own inputs, so the derivation chain is not circular.

Assumptions & free parameters 1 free parameters · 3 assumptions · 1 invented entities

The paper's central definition is a formula with one free parameter alpha. The main axioms are the meaningfulness of polar coordinates for persistence diagrams and the specific filtration choice. No new physical or mathematical entity is introduced beyond the metric formula itself.

free parameters (1)
  • alpha = 1.5 for sushi, unspecified for Irish
    Alpha controls the weight of the angular term in PPD. The paper uses a fixed value of alpha=1.5 for the sushi experiment and does not state the value used for the Irish experiment, nor does it test sensitivity to this parameter. The central claim of greater discriminative power depends on this choice.
assumptions (3)
  • domain assumption The filtration value f(i,j) = 1/(w_ij + epsilon) with epsilon=1e-6 yields persistence diagrams that meaningfully encode social choice structure.
    Section 5.2: this construction is arbitrary; the paper does not justify that this particular filtration weighting preserves the social choice semantics (dominance cycles, clusters) in a way that is stable or interpretable.
  • ad hoc to paper The angular coordinate theta = atan2(d,b) of a birth-death point is meaningful for comparing diagrams and invariant to the matching diagonal
    Section 4.2 and 4.3: this is the core novelty assumption. The paper provides no theoretical justification that the orientation relative to the origin captures relevant structure; it is an assumption that the authors make to motivate the metric.
  • standard math Stability of persistence diagrams under the filtration of a weighted directed graph via the one-skeleton lower-star filtration
    Section 5.2: the paper relies on standard persistent homology stability theorems for filtrations, but these typically apply to functions on a fixed complex; the paper does not verify that the directed graph filtration is a tame function on a finite complex with well-defined persistence.
invented entities (1)
  • Topological Social Choice (TSC)
    purpose: A conceptual label for applying topological methods to social choice systems
    The paper introduces this term as a new interdisciplinary field, but it is a label, not a new entity with falsifiable predictions. It does not provide an independent definition or benchmark.

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Cite this review

Pith. "Pith review of Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams." pith.science (2026). https://pith.science/paper/K5JZ3774

@misc{pith2026250714340,
  author       = {Pith},
  title        = {Pith review of: Topological Social Choice: Designing a Noise-Robust Polar Distance for Persistence Diagrams},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5JZ3774}},
  note         = {Machine review of arXiv:2507.14340}
}
read the original abstract

Topological Data Analysis (TDA) has emerged as a powerful framework for extracting robust and interpretable features from noisy high-dimensional data. In the context of Social Choice Theory, where preference profiles and collective decisions are geometrically rich yet sensitive to perturbations, TDA remains largely unexplored. This work introduces a novel conceptual bridge between these domains by proposing a new metric framework for persistence diagrams tailored to noisy preference data.We define a polar coordinate-based distance that captures both the magnitude and orientation of topological features in a smooth and differentiable manner. Our metric addresses key limitations of classical distances, such as bottleneck and Wasserstein, including instability under perturbation, lack of continuity, and incompatibility with gradient-based learning. The resulting formulation offers improved behavior in both theoretical and applied settings.To the best of our knowledge, this is the first study to systematically apply persistent homology to social choice systems, providing a mathematically grounded method for comparing topological summaries of voting structures and preference dynamics. We demonstrate the superiority of our approach through extensive experiments, including robustness tests and supervised learning tasks, and we propose a modular pipeline for building predictive models from online preference data. This work contributes a conceptually novel and computationally effective tool to the emerging interface of topology and decision theory, opening new directions in interpretable machine learning for political and economic systems.

Figures

Figures reproduced from arXiv: 2507.14340 by the authors.

Figure 1
Figure 1. Two persistence diagrams with equal bottleneck distance but distinct PPD. The angular configuration of [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Comparison of two persistence diagrams: Both contain features with the same birth–death distance [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. Comparison of two persistence diagrams: Both contain features with the same birth–death distance [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗

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Reference graph

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