{"id":"55516bd1-af0a-4e12-bcf4-a59cbb2b7138","arxiv_id":"2509.06577","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A neural network learns a reduced color ordering that approximates the Condorcet consensus of several lexicographic orderings, and this ordering defines morphological operators with lower irregularity on color images.","lead":"This paper trains a small neural network to learn a color ordering for image morphology by combining several existing orderings through Condorcet voting. The result is a consensus ordering that can be used in image filters and appears to produce more regular outputs than standard lexicographic or Borda orderings.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (12) minimizes the reverse of the intended Condorcet order, so as written the learned h* does not approximate the claimed Condorcet ordering.","rationale":"The reader's main concern was generalization and surrogate fidelity, which is legitimate but secondary. The sign error in Eqs. (7), (8), and (12) is a more fundamental internal inconsistency: the objective as written rewards the opposite of the majority pairwise relation, so the learned mapping cannot be the stated Condorcet approximation unless an unstated sign convention or implementation detail reverses it. This is not a disagreement with the consensus; it is a correctness risk in the derivation. The paper does contain useful machinery, and a sign flip with new experiments could make the method viable, but the manuscript in its current form does not establish the central claim. Hence the verdict moves from CONDITIONAL to REJECT, with the concrete test able to confirm whether the notebook shares the textual sign error or silently corrects it.","tokens_in":9895,"tokens_out":8843,"duration_ms":89282,"concrete_test":"Run the authors' provided Jupyter notebook on a reduced training set containing only black and white, with h1,h2,h3 defined by Eq. (15), and print h*(black) and h*(white) after training. If h*(white) < h*(black), the implemented loss reproduces the sign reversal of Eq. (12). Independently, compute the exact Kemeny-Young order for the four colors in Figure 5 using Eq. (7) with the corrected sign (maximizing sum delta_ij r_ij) and compare it with the h* rankings; a systematic reversal confirms the concern.","verdict_should_be":"REJECT","load_bearing_attack":"The sign convention in the optimization is internally inconsistent with the stated Condorcet goal. For any pair with delta_ij > 0 (meaning x_i is at most x_j in a majority of the voters), Eq. (8) and its soft version Eq. (12) assign positive cost to r_ij = [s_i <= s_j] and to sigma(s_j - s_i). Minimizing that term drives s_j - s_i toward -infinity, i.e., s_j < s_i, which makes x_j the smaller element and reverses the majority relation. Concretely, for the three lexicographic orderings of Eq. (15), black precedes white in all three, so delta_{black,white} = 1. Evaluating Eq. (12) on the two-color set {black, white} gives L(s_black=0, s_white=1) = sigma(1) ≈ 0.731, while L(s_black=1, s_white=0) = sigma(-1) ≈ 0.269. The minimizer therefore assigns h*(white) < h*(black), making white the least color, directly contradicting Section 4, which states that black is the least color and the Condorcet winner. The same issue appears in Eq. (7): the pair objective favors r_white,black = 1 over r_black,white = 1. Thus, unless the implementation silently flips the sign or the ordering direction, the trained h* approximates the reverse of the Condorcet consensus, and the central claim of the paper is not supported by the reported equations.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes to learn a scalar reduced ordering (an h-mapping) for vector-valued mathematical morphology by approximating the Condorcet consensus of a profile of lexicographic orderings. A multilayer perceptron maps RGB colors to real scores, and the parameters minimize the soft Kemeny-Young loss of Eq. (12); the resulting order is used with a look-up table to define morphological operators. Experiments are reported on 100 CIFAR images (with the next 100 as validation) and one BSD image, comparing lexicographic orderings, the Borda rule, and the learned h* in terms of loss curves, irregularity indices with Wilcoxon tests, and qualitative color rankings.","tokens_in":10285,"tokens_out":15724,"duration_ms":144187,"significance":"If the central claim were fully supported, the contribution would be a practical, learning-based way to aggregate multiple vector orderings into a reduced ordering for morphology, with the advantage of generalizing beyond a fixed finite color set and avoiding false colors through LUT-based total orders. The manuscript is reproducible (the Jupyter notebook is provided), the soft-loss formulation is clearly connected to Kemeny-Young voting, and the use of statistical tests for the irregularity comparison is a strength. However, because the paper never directly measures agreement with the Condorcet ordering and because the derivation in Section 3.2 contains sign inconsistencies, the main claim currently outruns the evidence.","major_comments":[{"comment":"The hard-to-soft derivation is internally inconsistent. With Eq. (6), δ_ij > 0 means x_i is less than or equal to x_j in a majority of the orderings. The hard objective (7), written as a minimization of Σ δ_ij r_ij, then prefers r_ij = 0 when δ_ij > 0, i.e., the reverse of the majority relation; Eq. (8) inherits this sign error. Moreover, Eq. (9) states that σ(s_j − s_i) replaces [s_i ≤ s_j], but σ(s_j − s_i) actually approximates [s_j ≤ s_i]. I checked the final soft loss (12): because the double sum contains both (i,j) and (j,i) with δ_ji = −δ_ij, the pair contribution is δ_ij(2σ(s_j − s_i) − 1); since σ is decreasing, for δ_ij > 0 the minimizer drives s_j > s_i, so Eq. (12) as written does implement the intended Condorcet direction. In the two-color example, the full loss (12) with (s_black, s_white) = (0,1) is σ(1) − σ(−1) ≈ −0.462, while the reversed assignment gives +0.462, so black is correctly favored as the smaller element. The stress-test concern that Eq. (12) reverses the Condorcet order therefore does not land for the full loss. The real problem is that Eqs. (7)–(10) are self-contradictory and must be repaired with a consistent sign convention, either by maximizing in (7)/(8) or by defining the logistic argument consistently with the step function it replaces.","section":"§3.2, Eqs. (6)–(10)"},{"comment":"The central claim that h* approximates the Condorcet ordering is never directly tested. The training and validation loss in Fig. 2 measures only the surrogate L(H,V_n); the boxplots and Wilcoxon tests in Fig. 3 measure the irregularity index, which is not an ordering-accuracy metric; and Fig. 5 is a hand-picked qualitative example. The paper should report the Kemeny score or a rank correlation such as Kendall's tau between the h* ranking and an exact Condorcet/Kemeny ranking on sampled subsets of the training and validation color sets, and also on synthetic profiles with a known Condorcet winner. Without such a measurement, the abstract's claim that the learned reduced ordering approximates the Condorcet ordering is unsupported.","section":"§4, Figs. 2, 3, and 5"},{"comment":"The relationship between the full objective (12) and the minibatch training is unspecified. The training set has n = 102,400 colors, so a full n×n matrix of δ_ij values is infeasible, but Eq. (13) defines θ* as the minimizer of L(H,V_n) over the whole V_n. If δ_ij is recomputed for each minibatch, the Condorcet target changes with the batch and no single h* minimizes a fixed, well-defined Condorcet problem; if δ is precomputed, the sampling scheme should be stated explicitly. The paper should specify exactly how δ enters the minibatch loss and how the validation loss in Fig. 2 relates to a fixed Condorcet instance.","section":"§3.3–§4, minibatch training"}],"minor_comments":[{"comment":"The word 'refered' should be spelled 'referred'.","section":"§3.3"},{"comment":"In the Borda example, the scores are given as B(x1) = 2/10, B(x2) = 7/10, and B(x1) = 6/10; the last should be B(x3), and 'x1 ≤B≤ x3' should read 'x1 ≤B x3'.","section":"§3.4"},{"comment":"The caption says 'All the openings have been computed' but the figure and the surrounding text describe closings; the caption should say 'closings'.","section":"Figure 4 caption"},{"comment":"The sentence 'the optimization problem outlined in (10), with the loss function defined by (12)' is confusing because (10) is the score-level soft Condorcet problem, while the network training problem is (13); please rephrase to distinguish the two.","section":"§4, training paragraph"},{"comment":"The terminology 'Condorcet winner' is inverted relative to standard social-choice usage, where the winner is usually the most preferred candidate rather than the least element of a 'less than or equal' order; one clarifying sentence about this convention would help readers.","section":"§3.1"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern that Eq. (12) reverses the intended Condorcet order does not survive a check of the full double sum, so reject on that ground is not justified. The more serious issue is the absence of any direct quantitative evaluation of the approximation to the Condorcet ordering, together with the sign inconsistency in the derivation in Section 3.2; both are fixable within the manuscript's scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper's central idea is worth a look, but the objective as written learns the opposite of the Condorcet order. That's the first thing you need to know.\n\nThe setup is nice. They take the soft Condorcet optimization of Lanctot et al., which ranks a fixed set of candidates, and turn it into a learned reduced ordering for vector-valued morphology, so it can rank colors never seen in training. That is a genuine extension and a sensible direction for a niche area. The writing is clear, the code is on GitHub, and they evaluate on CIFAR and one BSD image with a standard irregularity index.\n\nThe problem is the sign of the loss. In Eq. (6), delta_ij > 0 means a majority of the h-mappings put x_i at most x_j. In Eq. (7) and (8) they minimize sum delta_ij * [s_i <= s_j]. For a pair with delta_ij > 0, that term is small when [s_i <= s_j] = 0, i.e., when s_j < s_i, which is the opposite of what the majority says. Eq. (12) has the same issue with the logistic; minimizing it drives the score of the majority-smaller element above the score of the majority-larger element. Concretely, for black and white with the three lexicographic orders in Eq. (15), all three put black before white, so delta_black,white = 1. The loss prefers h(white) < h(black). But Section 4 says black is the least color. So the equations as written cannot produce the reported behavior unless the code silently flips a sign. A referee should check the code, but the paper's own math does not support the headline claim.\n\nEven with the sign fixed, the evaluation is thin: no direct measurement of how close the learned ordering is to an exact Condorcet ordering on held-out colors, no Kemeny score, just one profile of three lexicographic orders and one irregularity index. Those are addressable, though.\n\nSo, net: interesting idea, but a load-bearing error in the submitted equations. I wouldn't cite it as-is, and I wouldn't rush to build on it until the sign is resolved. It does deserve peer review—a serious referee should get the code and run the two-color experiment I described. If the implementation is correct and the equations are just misstated, this could become a decent paper in vector-valued morphology.","headline":"The paper has a promising idea but the loss function as written reverses the Condorcet ordering, which is a load-bearing error.","tokens_in":10753,"tokens_out":9924,"would_cite":false,"duration_ms":82130,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68U10","91B12"],"pacs":[],"model":"deepseek-v4-flash","headline":"A neural network learns a scalar map that approximates the Condorcet consensus of several vector orderings; the map defines color-image morphology with fewer irregularities than lexicographic or Borda orderings.","keywords":["vector-valued mathematical morphology","reduced ordering","Condorcet ordering","Kemeny–Young method","soft Condorcet optimization","multilayer perceptron","color image processing","social choice theory"],"falsifier":"Take a small set of held-out colors (fewer than about 20, small enough that the NP-hard problem is tractable), solve the Kemeny–Young problem (7) exactly by exhaustive search over all total orders, and compare that true Condorcet ordering with the ordering induced by the trained $h^*$. If $h^*$ violates a pairwise majority margin — that is, $\\delta_{ij} > 0$ yet $h^*(x_i) > h^*(x_j)$ — for a nontrivial fraction of pairs, or disagrees with the exact consensus ranking on a substantial fraction of adjacencies, the approximation claim on unseen colors is refuted.","tokens_in":9734,"feed_emoji":"🗳️","tokens_out":21990,"duration_ms":157312,"temperature":0.7,"pith_summary":"Choosing an ordering for vector-valued mathematical morphology — the step that decides which colors count as 'less' or 'greater' in a dilation or erosion — has no canonical answer. This paper treats the choice as a voting problem: each candidate ordering is a voter, and the target is the Condorcet consensus, the ranking that beats every alternative in pairwise majority comparisons. Because finding the exact Condorcet ordering is NP-hard, the authors minimize a smooth surrogate loss instead, training a small neural network to map every RGB color to a single real score. The paper claims that the resulting scalar mapping reproduces the Condorcet ranking of the training colors, generalizes to images outside the training set, and — combined with a look-up table — defines morphological operators that never introduce colors absent from the image and produce fewer irregularities than three lexicographic orderings or the Borda rule. If correct, this replaces an ad hoc design choice with a principled, data-driven consensus ordering.","feed_headline":"Learned color ranking makes image filters smoother, not false-colored","feed_subtitle":"Trained to aggregate pairwise votes among rival orderings, it approximates the Condorcet consensus on image colors.","key_machinery":"The central object is the reduced ordering: a surjective scalar mapping $h: V \\to \\mathbb{R}$ that induces the pre-order $x \\leq_h y \\iff h(x) \\leq h(y)$, which, combined with a look-up table, becomes a total order on the finite set of colors appearing in an image and therefore supplies the suprema and infima that dilations and erosions need. The argument is carried by the soft Condorcet optimization loss introduced in [12]: the hard Kemeny–Young problem of minimizing $\\sum_{i,j} \\delta_{ij} r_{ij}$ over all total-order matrices $R = (r_{ij})$ — NP-hard, with transitivity enforced by $r_{ij} + r_{jk} - r_{ik} \\leq 1$ — is smoothed by replacing the discontinuous step function $[s_i \\leq s_j]$ with the logistic $\\sigma(s_j - s_i)$, so the consensus ordering becomes the output of a small multilayer perceptron (3-64-1, ReLU hidden layer) trained by gradient descent. The pairwise vote margins $\\delta_{ij}$ computed once from the voter family $H$ are the fixed data; the network learns a single scalar ranking function that reproduces those margins and extends beyond the training colors.","core_discovery":"For a finite set of colors $V_n$ and a family $H = \\{h_1, \\dots, h_m\\}$ of reduced mappings (the 'voters'), the paper defines the Condorcet $h^*$-mapping as $h(\\theta^*)$, where $\\theta^*$ minimizes the soft Kemeny–Young loss $$L(H,V_n)(\\$\\theta$) = \\sum_{i=1}^n \\sum_{j=1}^n \\delta_{ij}\\,\\$\\sigma$\\!\\left(h(\\$\\theta$)(x_j) - h(\\$\\theta$)(x_i)\\right),$$ with $\\delta_{ij}$ the average pairwise vote margin between colors $x_i$ and $x_j$ computed over $H$, and $\\sigma(u) = 1/(1+e^{u/\\tau})$ the logistic surrogate for the step function (here $\\tau = 1$). The paper claims that this trained scalar mapping orders the colors the way the Condorcet consensus would — the unanimous extremes fall in place (black least, white greatest), with contested intermediate colors placed by the majority margins — and that, because $h^*$ is a function on the entire color cube, the same mapping defines morphological operators for new images without re-solving the optimization. On CIFAR and BSD images, openings and closings built from $h^*$ plus a look-up table show lower and less variable global irregularity than the three lexicographic orderings and the Borda rule, a difference the authors support with a Wilcoxon signed-rank test at 99% confidence.","pith_inferences":["The paper fixes the logistic temperature at $\\tau = 1$ and the network at 3-64-1 without ablating either; a natural extension is to test how closely the learned ranking tracks the exact Kemeny–Young ordering as $\\tau$ shrinks and the network grows, separating the surrogate's approximation error from the network's.","Because the loss sees only pairwise margins, the voter family itself could be learned or made image-dependent, so the consensus ordering would adapt to image content instead of being fixed by three hand-chosen lexicographic rules.","The reported irregularity advantage rests on one specific index; a natural test would be whether the learned ordering also improves downstream tasks such as noise filtering or border detection.","The hyperspectral and abundance-simplex extension the authors flag is direct because the loss is dimension-agnostic; the open question is whether the advantage survives heterogeneous voters rather than three permutations of the same lexicographic rule."],"forward_implications":["A single trained scalar mapping can stand in for the whole voter family: the learned $h^*$ reproduces the consensus ranking of the three lexicographic orderings on the training colors, including the unanimous extremes — black least, white greatest.","Morphological operators built from the learned reduced ordering plus a look-up table introduce no 'false colors' and show lower, less variable irregularity than the three lexicographic orderings and the Borda rule, on both training and held-out images.","Because $h^*$ is defined on the entire color cube, images outside the training set are processed with the same operators without re-solving the optimization or recomputing any consensus.","The NP-hard Kemeny–Young consensus problem is moved into an offline training step: deployment of the voting machinery costs one forward pass and a sort per image.","The same loss aggregates any family of reduced mappings — distance-based, marginal, or stochastic-permutation orderings — so the consensus criterion is not tied to the three lexicographic voters used in the experiments."],"supporting_citations":[{"why":"Supplies the soft Condorcet optimization method — the logistic surrogate and gradient-descent training — that the loss in Eq. (12) is built on.","marker":"[12]"},{"why":"Provides the linear-programming formulation and the transitivity/antisymmetry constraints of the Condorcet ordering problem restated in Eq. (7).","marker":"[14]"},{"why":"Establishes the Kemeny–Young method as the Condorcet-consistent consensus rule that the paper approximates.","marker":"[23]"},{"why":"Formalizes the social-choice setting — profile, voters, Condorcet winner — that the paper reinterprets as a family of reduced mappings.","marker":"[24]"},{"why":"Defines reduced orderings combined with look-up tables for vector-valued morphology, the mechanism that turns the learned mapping into operators without false colors.","marker":"[20]"},{"why":"Supplies the CIFAR-10 images whose colors form the 102,400-color training and validation sets used to train and assess the network.","marker":"[10]"},{"why":"Defines the global irregularity index used to compare operators built from the lexicographic orderings, the Borda rule, and the learned mapping.","marker":"[19]"},{"why":"Provides the Condorcet principle and the Borda rule as competing consensus methods, the basis of the comparison in Section 4.","marker":"[4]"}],"fun_headline_variants":["Machine learning approximates Condorcet for color filtering","Vote-based ranking yields smoother color morphology","Learned consensus ordering improves image filters","Condorcet-inspired ML ranking for cleaner color images"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that minimizing the smoothed surrogate loss on batches of training colors yields a scalar ordering that stays close to the true Condorcet ordering for colors and images the network never saw; no direct comparison against an exact Condorcet ordering is reported.","fun_headline_variants_meta":{"raw":{"variants":["Machine learning approximates Condorcet for color filtering","Vote-based ranking yields smoother color morphology","Learned consensus ordering improves image filters","Condorcet-inspired ML ranking for cleaner color images"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000299,"raw_usage":{"total_tokens":1749,"prompt_tokens":984,"completion_tokens":765,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":600,"completion_tokens_details":{"reasoning_tokens":707}},"tokens_in":600,"tokens_out":765,"duration_ms":7467,"temperature":1.0,"reasoning_tokens":707,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T16:15:46.446693+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a small set of held-out colors (fewer than about 20, small enough that the NP-hard problem is tractable), solve the Kemeny–Young problem (7) exactly by exhaustive search over all total orders, and compare that true Condorcet ordering with the ordering induced by the trained $h^*$. If $h^*$ violates a pairwise majority margin — that is, $\\delta_{ij} > 0$ yet $h^*(x_i) > h^*(x_j)$ — for a nontrivial fraction of pairs, or disagrees with the exact consensus ranking on a substantial fraction of adjacencies, the approximation claim on unseen colors is refuted.","supporting_citations":[{"cited_title":"Soft Condorcet Optimization for Ranking of General Agents","cited_arxiv_id":"2411.00119","evidence_quote":"Supplies the soft Condorcet optimization method — the logistic surrogate and gradient-descent training — that the loss in Eq. (12) is built on."},{"cited_title":"Revue de statistique appliquée30(2), 21–44 (1982)","cited_arxiv_id":null,"evidence_quote":"Provides the linear-programming formulation and the transitivity/antisymmetry constraints of the Condorcet ordering problem restated in Eq. (7)."},{"cited_title":"SIAM Journal on Applied Mathematics35(2), 285–300 (9 1978)","cited_arxiv_id":null,"evidence_quote":"Establishes the Kemeny–Young method as the Condorcet-consistent consensus rule that the paper approximates."},{"cited_title":"In: Handbook of Computa- tional Social Choice, pp","cited_arxiv_id":null,"evidence_quote":"Formalizes the social-choice setting — profile, voters, Condorcet winner — that the paper reinterprets as a family of reduced mappings."},{"cited_title":"In: Celebi, M.E., Smolka, B","cited_arxiv_id":null,"evidence_quote":"Defines reduced orderings combined with look-up tables for vector-valued morphology, the mechanism that turns the learned mapping into operators without false colors."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the CIFAR-10 images whose colors form the 102,400-color training and validation sets used to train and assess the network."},{"cited_title":"Journal of Mathematical Imaging and Vision64(7), 754–770 (4 2022)","cited_arxiv_id":null,"evidence_quote":"Defines the global irregularity index used to compare operators built from the lexicographic orderings, the Borda rule, and the learned mapping."},{"cited_title":"(eds.): Handbook of Computational Social Choice","cited_arxiv_id":null,"evidence_quote":"Provides the Condorcet principle and the Borda rule as competing consensus methods, the basis of the comparison in Section 4."}],"review_version":1}