Approximating Condorcet Ordering for Vector-valued Mathematical Morphology

Gustavo Jesus Angulo; Joao Batista Florindo; Marcos Eduardo Valle; Santiago Velasco-Forero

REVIEW 3 major objections 5 minor 24 references

Approximating Condorcet Ordering for Vector-valued Mathematical Morphology

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict The paper has a promising idea but the loss function as written reverses the Condorcet ordering, which is a load-bearing error. read the letter →

arxiv 2509.06577 v1 pith:Q763NB7Q submitted 2025-09-08 cs.CV cs.LGcs.NE

Marcos Eduardo Valle , Santiago Velasco-Forero , Joao Batista Florindo , Gustavo Jesus Angulo This is my paper · ORCID

classification cs.CVcs.LGcs.NE MSC 68U1091B12

keywords vector-valuedmathematicalmorphologyreducedorderingCondorcetKemeny–Youngmethodsoftoptimizationmultilayerperceptroncolorimageprocessingsocialchoicetheory

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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the 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.

The 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.

The 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.

Even 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.

So, 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.

Referee Report

3 major / 5 minor

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.

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 (3)

[§3.2, Eqs. (6)–(10)] 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.
[§4, Figs. 2, 3, and 5] 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.
[§3.3–§4, minibatch training] 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.

minor comments (5)

[§3.3] The word 'refered' should be spelled 'referred'.
[§3.4] 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'.
[Figure 4 caption] The caption says 'All the openings have been computed' but the figure and the surrounding text describe closings; the caption should say 'closings'.
[§4, training paragraph] 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.
[§3.1] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the Condorcet objective is an external target, and self-citations are supporting tools only.

full rationale

After walking the derivation chain, I find no circular step that reduces the paper's central claims to its inputs. Equations (12)-(13) define h* as the minimizer of a soft surrogate for the Kemeny objective in Eqs. (7)-(8); the Condorcet consensus is an external social-choice construct, not a quantity fitted from h*'s outputs and then reported as a prediction. The training-set agreement with the Condorcet ranking is a sanity check of the optimizer, and the paper's main evidence for generalization is the validation loss (Figure 2) and the irregularity-index comparison (Figure 3), neither of which appears in the training loss. The authors' prior work is used for supporting tools — the reduced-ordering/LUT construction [20] and the irregularity index [19] — but these are not premises that force the learned ordering, and no uniqueness theorem or ansatz is imported from self-citations. A separate sign-convention issue in Eqs. (7)-(12) relative to delta_ij would be a correctness problem if valid, but it is not a circularity problem. Overall, the derivation is self-contained against external benchmarks, with only minor self-citations that are not load-bearing.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical or conceptual entities are introduced; the learned h*-mapping is a parameterized function trained on data, not a postulated object. The free parameters are the soft-step temperature, the network and training choices, and the fitted network weights. The axioms are mostly standard background assumptions, with the generalization assumption and the hand-picked profile H being the least supported.

free parameters (3)

τ (logistic steepness) = 1
Appears in Eq. (9) as the soft step steepness; set to 1 for all experiments with no sensitivity analysis. The learned ordering and all downstream operators depend on this choice.
MLP architecture and training schedule = 3-64-1, 100 epochs, batch size 1024, Adam defaults
Hand-chosen for the experiments; no ablation or justification is given, and the central generalization claim depends on these choices being adequate.
MLP weights (3-64-1 network) = Learned from the 102,400 training colors
The central claim relies on this particular fitted function; no constraints, regularization, or architecture justification are given.

assumptions (5)

domain assumption V is a complete lattice and reduced orderings combined with look-up tables define valid morphological operators.
Invoked in Section 2 to justify dilation and erosion and the no-false-color property; standard in mathematical morphology but assumed rather than proved here.
domain assumption The soft logistic surrogate σ in Eq. (9) faithfully approximates the 0-1 Kemeny objective so that minimizing Eq. (12) yields a near-Condorcet ordering.
Borrowed from Lanctot et al. [12]; the paper adopts τ=1 and does not verify the surrogate's fidelity for the learned h*.
ad hoc to paper The family H = {h1, h2, h3} of lexicographic orderings is a meaningful profile whose Condorcet consensus is a desirable vector ordering.
The three h-mappings in Eq. (15) are chosen 'for illustrative purposes'; the entire evaluation is relative to this hand-picked profile.
domain assumption A mapping trained on the first 100 CIFAR-10 images generalizes to other color images, including BSD.
Stated in Section 3.3 as generalization capability and tested on CIFAR validation and one BSD image; no theoretical guarantee or broad empirical support.
domain assumption The irregularity index Φ_g^1 from [19] is a valid proxy for morphological operator quality.
Used as the main quantitative evidence in Section 4; it is a metric introduced by the same research group.

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Cite this review

Pith. "Pith review of Approximating Condorcet Ordering for Vector-valued Mathematical Morphology." pith.science (2026). https://pith.science/paper/Q763NB7Q

@misc{pith2026250906577,
  author       = {Pith},
  title        = {Pith review of: Approximating Condorcet Ordering for Vector-valued Mathematical Morphology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q763NB7Q}},
  note         = {Machine review of arXiv:2509.06577}
}

read the original abstract

Mathematical morphology provides a nonlinear framework for image and spatial data processing and analysis. Although there have been many successful applications of mathematical morphology to vector-valued images, such as color and hyperspectral images, there is still no consensus on the most suitable vector ordering for constructing morphological operators. This paper addresses this issue by examining a reduced ordering approximating the Condorcet ranking derived from a set of vector orderings. Inspired by voting problems, the Condorcet ordering ranks elements from most to least voted, with voters representing different orderings. In this paper, we develop a machine learning approach that learns a reduced ordering that approximates the Condorcet ordering. Preliminary computational experiments confirm the effectiveness of learning the reduced mapping to define vector-valued morphological operators for color images.

Figures

Figures reproduced from arXiv: 2509.06577 by the authors.

**Figure 1.** Illustrative images from the training set T (first column) and the corresponding opened images obtained using the reduced mappings h1, h2, h3, the Borda rule B, and the Condorcet h ∗ -mapping [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗

**Figure 2.** The loss over epochs during the training of the Condorcet h ∗ -mapping. values, Φ g 1 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Boxplot of the global irregularity values and the Hasse diagram of pair-wise Wilcoxon signed-rank test with confidence level at 99%. cyan, the second largest color in the lexicographical G − B − R ordering, as the second largest color in the Condorcet ordering. In general, the ranking of colors yielded by h ∗ agrees with the Condorcet order derived from the three lexicographic orderings, confirming the proposed appr… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: First row: A color image alongside its closing processed using Borda rule and the Condorcet h ∗ -mapping. Second row: Closing by Lexicographic orderings h1 (R−G−B), h2 (G − B − R), and h3 (B − R − G). 2. Aptoula, E., Lefèvre, S.: A Comparative Study on Multivariate Mat…

**Figure 5.** Figure 5: Illustrative example of the ranking obtained using the lexicographic orderings and the Condorcet h ∗ -mapping. 13. Lézoray, O.: Mathematical morphology based on stochastic permutation orderings. Mathematical Morphology - Theory and Applications 5(1), 43–69 (1 2021) 14.…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Computer vision and image understanding62, 326–346 (1995)

Goutsias, J., Heijmans, H.J.A.M., Sivakumar, K.: Morphological Operators for Image Sequences. Computer vision and image understanding62, 326–346 (1995)

work page 1995