Families of Optimal Transport Kernels for Cell Complexes

Rahul Khorana

REVIEW 6 major objections 5 minor 25 references

Families of Optimal Transport Kernels for Cell Complexes

T0 review · 6 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper derives a closed-form Wasserstein-2 distance between CW complexes from their Hodge Laplacians and builds optimal-transport kernels and a Gaussian process on top.

desk verdict A natural but unproven extension of graph OT to cell complexes, undermined by a central formula that is off by a square root and an experiment with no baselines. read the letter →

arxiv 2507.16569 v1 pith:PSJVGBVU submitted 2025-07-22 cs.LG stat.ML

Rahul Khorana This is my paper · ORCID

classification cs.LGstat.ML

keywords optimaltransportWassersteindistanceCWcomplexesHodgeLaplacianGaussianprocessesFusedGromov-Wassersteinkernelmethodscell

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

Cell complexes are a flexible topological representation for learning, but few machine-learning methods operate on them directly. This paper seeks to change that by defining a structural distance: it models the signals on a CW complex as a centered Gaussian whose covariance is the Moore-Penrose pseudoinverse of the complex's Hodge Laplacian, then derives the Wasserstein-2 distance between two such distributions in closed form. That formula turns the Hodge Laplacians of two complexes into a distance and an optimal transport map, giving a direct way to compare complexes of the same topological dimension. The paper then extends the Fused Gromov-Wasserstein distance to CW complexes so cell features and structure enter together, and builds exponential kernels from both distances, using a spectral-truncation result to guarantee positive definiteness so a Gaussian process can be trained. A reader would care because this is a principled bridge from cell-complex geometry to kernel-based learning.

What carries the argument

The object carrying the argument is the weighted Hodge Laplacian of a CW complex, $\Delta_k = B_k^{\top} W_{k-1}^{-1} B_k W_k + W_k^{-1} B_{k+1} W_{k+1} B_{k+1}^{\top}$, built from boundary matrices $B_k$ and diagonal cell-weight matrices $W_k$. The argument works by taking the Moore-Penrose pseudoinverse of this matrix as the covariance of a centered Gaussian signal distribution, so the standard closed-form Wasserstein-2 identity for Gaussian measures applies verbatim. A second piece of machinery is the spectral-truncation positive-definiteness rule: for any symmetric zero-diagonal distance on a finite dataset there is a bandwidth threshold below which the exponential Gram matrix is positive semidefinite, and truncating its spectral decomposition gives finite Wasserstein features usable in a Gaussian process.

What would settle it

Take two CW complexes, draw signals from a known non-Gaussian process on each, estimate the empirical Wasserstein-2 distance between the samples, and compare it with the value of Eq. (14) computed from the Hodge Laplacians; disagreement beyond sampling error would disprove the Gaussian-pseudoinverse modeling claim.

Watch

Extended reading notes

Core claim

The central claim is Definition 3.4: represent a finite $n$-dimensional CW complex $X_i$ by the measure $\mu_{X_i} = \mathcal{N}(0,\Delta_{X_i}^{\dagger})$, where $\Delta_{X_i}$ is the weighted Hodge Laplacian and $\Delta_{X_i}^{\dagger}$ is its Moore-Penrose pseudoinverse. For two such complexes, the paper asserts the Wasserstein-2 distance between these signal distributions has the closed form $$W_2(\mu_{X_1},\nu_{X_2}) = \operatorname{Tr}(\Delta_{X_1}^{\dagger} + \Delta_{X_2}^{\dagger}) - 2\operatorname{Tr}\left(\sqrt{(\Delta_{X_1}^{\dagger})^{1/2}\,\Delta_{X_2}^{\dagger}\,(\Delta_{X_1}^{\dagger})^{1/2}}\right),$$ with the optimal transport map also given explicitly in terms of the same matrices. The paper extends this to a Fused Gromov-Wasserstein distance (Definition 3.5) that mixes cell-weight feature distances with Hodge-Laplacian structure tensors, then defines exponential kernels from both distances and enforces positive definiteness by choosing a bandwidth below a guaranteed threshold and truncating the spectral decomposition. It reports that the feature-plus-structure kernel achieves lower test loss than the pure Wasserstein kernel in a Gaussian-process experiment on a synthetic dataset of CW complexes.

Load-bearing premise

The whole construction assumes that signals on a CW complex are normally distributed with covariance equal to the Moore-Penrose pseudoinverse of the Hodge Laplacian; if that is not the right model for the signals, the closed-form distance and the kernels do not describe the actual signal distributions.

Editorial extensions

If this is right

For any two finite CW complexes of the same topological dimension, the formula gives a structural distance and an explicit optimal transport map that moves one signal distribution onto the other.
The Fused Gromov-Wasserstein extension lets a kernel blend cell-weight features with Hodge-Laplacian structure, and the paper's experiment reports a lower test loss for this kernel than for the pure Wasserstein kernel.
The bandwidth-threshold result guarantees that the exponential kernels can be made positive definite on any finite dataset, so they can drive a Gaussian process over CW complexes.
Because the distance is computed directly from Hodge Laplacians, CW complexes need not share cell counts or attaching patterns; only the dimension of the Hodge-Laplacian matrices must match for the Wasserstein comparison.

Reading between the lines

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

Editorial inference: if real cell signals are not Gaussian with the Hodge pseudoinverse as covariance, the closed form still serves as a spectral pseudo-metric on complexes, and Eq. (14) can be reinterpreted as a kernel on Hodge spectra rather than on signal distributions.
Editorial inference: because the distance depends only on the Hodge Laplacians, two complexes with unitarily equivalent Hodge Laplacians are indistinguishable under this measure; augmenting it with orientation or attaching-map data would be a natural next step.
Editorial inference: the same positive-definiteness enforcement applies to any symmetric zero-diagonal distance, so the kernel construction could be reused with other transport costs, including approximate Sinkhorn costs, without changing the spectral-truncation step.
Editorial inference: applying the framework to graph benchmarks by treating graphs as one-dimensional CW complexes would give a direct comparison against graph Wasserstein kernels and test whether the extra cell structure helps.

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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

Rahul Khorana's paper adapts two known graph-OT constructions to cell complexes by replacing the graph Laplacian with the Hodge Laplacian. The topic is timely and the writing is clear; the paper is the first I know of to write down OT kernels directly on CW complexes. That is the good part.

The central problem is in Definition 3.4. Definition 1.5 correctly defines W_p as the p-th root of the transport cost. But Eq. (13) drops the root, and Eq. (14) states W2 equals the trace expression that is actually the squared 2-Wasserstein distance for centered Gaussians. So the paper's main distance is off by a square root. The kernels in Eqs. (18)–(19) and the losses in Table 1 are computed from this quantity, so the experimental numbers don't correspond to a Wasserstein-2 distance as defined. This is a load-bearing flaw, not a typo.

Beyond that, the paper says it 'derives' Eq. (14) but gives no derivation. The formula is a direct substitution into the Gaussian Wasserstein identity, which is fine, but it should be cited and proved. Definition 3.5 has a couple of inconsistencies too: the coupling set is written as Π(µ,µ), and the feature-distance term in Eq. (17) uses d(w^k_i, w^k_j) without distinguishing the two complexes.

The embedding theorem (3.1) is true for finite CW complexes, but the proof confuses homotopy equivalence with embedding and invokes a regular neighborhood without establishing the manifold setup. The Menger–Nöbeling remark is the cleaner route and could be the whole proof.

The experiment is a placeholder: 1000 samples, 15 epochs, no baselines, no error bars, no code. The test-loss comparison between the two kernels is not evidence without uncertainty quantification or a baseline. The limitation section admits this, which is honest but doesn't make the evaluation convincing.

Net: the core idea is reasonable, but as written the central formula is wrong, the derivation is missing, and the experiments don't validate anything. I would not cite it in its current form. With a corrected formula, a proper derivation, and a real experiment, it could become a small but useful contribution. I'd send it to reviewers rather than desk-reject it, because an expert can confirm the fix and push for the missing evaluation. If the policy is to reject without review for papers with a central mathematical error, that's also defensible, but I think the idea deserves referee time.

Referee Report

6 major / 5 minor

Summary. The paper proposes to compare CW complexes by treating signals on each complex as a centered Gaussian with covariance equal to the Moore-Penrose pseudoinverse of the Hodge Laplacian. It claims to derive a closed-form Wasserstein-2 distance between such signal distributions, introduces a Fused Gromov-Wasserstein distance for CW complexes, and defines exponential kernels on the space of probability measures over CW complexes. The empirical section fits Gaussian processes with these kernels and reports that the FGW kernel achieves lower test loss than the Wasserstein kernel. The paper explicitly acknowledges the absence of baseline comparisons.

Significance. The idea of lifting graph optimal transport, as in Petric Maretic et al., to cell complexes via Hodge Laplacians is natural and potentially useful for geometric machine learning. If the central formula were correct, it would provide a closed-form structural dissimilarity for cell complexes and a route to kernel-based Gaussian processes. The paper is also transparent about its main limitation, namely that no baselines are provided. However, the theoretical core currently contains a misstated central formula, an incomplete embedding construction, and an ill-posed FGW definition, so the contribution is not yet established.

major comments (6)

[§3.3, Eqs. (13)–(14)] Definition 1.5 (Eq. 3) defines W_p as the p-th root of the transport cost. Equation (13) drops this root for p=2, and Eq. (14) reproduces the standard Dowson–Landau formula for W_2^2, not for W_2. Because Eqs. (18)–(19) and Table 1 use distances computed from Eq. (14), all reported quantities are squared Wasserstein distances, not Wasserstein-2 distances. The paper must either explicitly define a squared-distance object and use it consistently, or take the square root in Eq. (14) and propagate the change through the kernels and experiments.
[§3.3, Definition 3.4] The two Gaussian measures are defined on Euclidean spaces whose dimensions equal the numbers of cells, but the trace formula in Eq. (14) is only meaningful when the two covariance matrices have the same size. The paper introduces a Polish embedding X, yet never explains how N(0,∆†_{X1}) and N(0,∆†_{X2}) are realized as measures on the same space when X1 and X2 have different numbers of cells. Section 5 later restricts to equal-dimension Hodge Laplacians, narrowing the stated generality. A precise construction of a common ambient space, and an extension of Eq. (14) to unequal dimensions, are required.
[§3.1, Theorem 3.1] The proof does not establish an embedding of X. It shows that X is homotopy equivalent to a simplicial complex S, that S embeds linearly in R^k, and that a regular neighbourhood M of that embedding is homotopy equivalent to X. The smooth manifold M is then embedded in Euclidean space, but the resulting subset is not an embedding of X. The later phrase 'let X1 and X2 correspond to the polish space embeddings of X1 and X2' therefore conflates X with a homotopy-equivalent manifold. Since any finite CW complex is a compact metric space and hence Polish, a direct proof is possible, but the current argument does not justify Definitions 3.3–3.5.
[§3.4, Definition 3.5] The CW-FGW distance is not well defined. It sets µ = N(0,∆†_{X1}) and ν = N(0,∆†_{X2}), but the minimization in Eq. (16) is over Π(µ,µ) rather than Π(µ,ν), and Eq. (17) expands the cost using discrete indices π_{ij} with sums over i,j,k,l, as in the discrete graph FGW of Definition 2.2. With Gaussian continuous measures, the meaning of these couplings and of the tensor L(∆_{X1},∆_{X2}) is undefined. Moreover, the matrix M_{k,X1,X2} is only k×k if both complexes have the same number of k-cells, which is not assumed anywhere in the definition.
[§4.3, Definition 4.1] The spectral truncation in Definition 4.1 produces a Nyström-style feature map defined relative to a fixed finite set X. It does not by itself define a positive definite kernel function on the entire input space, so the claim that this 'enables us to define a Gaussian process over CW-complexes' is not justified. The authors should either prove that the resulting approximate kernel φ(x)^T φ(y) is positive definite for all inputs and state that this approximate kernel is what is used, or weaken the Gaussian-process claim.
[§6 and Table 1] The paper explicitly acknowledges in Section 6 that no baseline comparisons are provided. Given that the only empirical result is that kFGW outperforms kW on a single dataset, the claim that the FGW kernel is 'more expressive and efficient' is not supported by evidence against existing graph or topological learning methods. If the paper is intended as a theoretical contribution, the empirical section should be framed as a proof of concept; if it claims an empirical advantage, baselines are needed.

minor comments (5)

[§4.1, Eq. (18)] Equation (18) uses W_p without squaring in the exponent, while Eq. (20) defines a squared-exponential kernel with d^2; this notation should be aligned.
[§3.4, Eq. (16)] The coupling set in Eq. (16) is written as Π(µ,µ), which appears to be a typo for Π(µ,ν); the surrounding text should be corrected.
[Algorithm 1] The parameter update is written as θ_{i+1} ← -θ_i - γ∇J(θ), which is not gradient descent; it should be θ_{i+1} ← θ_i - γ∇J(θ).
[Definition 3.4] The phrase 'countable product of Polish spaces' is incorrect for a product of two spaces; the property invoked is for finite or countable products, but here only a finite product is needed.
[Theorem 3.1 proof] The reference to 'Whitney (2012)' for the Whitney embedding theorem is nonstandard; a standard textbook citation such as Lee (2012) would be more appropriate and is already in the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Eq. 14 directly evaluates a defined Gaussian Wasserstein distance using a standard external closed form.

full rationale

Definition 3.4 fixes the signal model: measures on CW complexes are taken to be centered Gaussians with covariance equal to the Moore-Penrose pseudoinverse of the Hodge Laplacian, and the CW Wasserstein distance is then W2 between these Gaussians. Equation 14 is obtained by substituting Δ† into the classical closed form for the quadratic Wasserstein distance between centered Gaussians, which the paper quotes in Definition 2.1 from Petric Maretic et al. (2019). This is a direct application of an external, standard result, not a fitted parameter renamed as a prediction, and no step defines the distance in terms of the trace expression it is supposed to derive. The kernels (Eqs. 18-19) use the distance as an input, and the positive-definiteness correction in Section 4.3 is an external result from De Plaen et al. (2020). The experimental section fits a Gaussian process by marginal likelihood on held-out data; the distance formula itself is not fit. The paper contains no self-citations, so there is no self-citation chain carrying the argument. The paper's own stated limitation (Sec. 6: 'We do not provide baseline comparisons...') concerns empirical validation, not the derivation's independence. A non-circularity concern should be noted separately: under the paper's own Definition 1.5 (Wp = T_p^{1/p}), Eqs. 13-14 state W2 = integral/trace, which is actually W2 squared; that is an internal mathematical inconsistency, not a circular reduction. Hence the derivation chain is self-contained and the central claim does not reduce to its inputs.

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

The framework's central formulas are reused from graph OT literature with matrix substitution, and the Gaussian signal model is introduced to make the closed-form identities applicable.

free parameters (3)

kernel bandwidth σ
Appears in Equations (18) and (19) as a hyperparameter; the paper does not specify how it is selected in the experiments.
FGW tradeoff α
Parameter in the CW-FGW distance (Definition 3.5) controlling the balance between feature and structure costs; no fitting procedure is described.
eigenvalue truncation count ℓ
Used in Definition 4.1 to construct a finite-dimensional feature map; the choice of ℓ is not discussed.

assumptions (3)

standard math Closed-form Wasserstein-2 distance between centered Gaussian measures
Invoked implicitly in Definition 3.4; this identity is a known result from optimal transport but is not proven in the paper.
domain assumption Finite CW complexes are embeddable as Polish spaces
The proof of Theorem 3.1 is incomplete; it conflates homotopy equivalence with embedding and relies on unproven regular neighborhood claims.
ad hoc to paper Signal distributions are Gaussian with covariance equal to the pseudoinverse of the Hodge Laplacian
This modeling choice appears in Definitions 3.4 and 3.5 without justification or validation, and it makes the closed-form formulas available.

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Cite this review

Pith. "Pith review of Families of Optimal Transport Kernels for Cell Complexes." pith.science (2026). https://pith.science/paper/PSJVGBVU

@misc{pith2026250716569,
  author       = {Pith},
  title        = {Pith review of: Families of Optimal Transport Kernels for Cell Complexes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PSJVGBVU}},
  note         = {Machine review of arXiv:2507.16569}
}

read the original abstract

Recent advances have discussed cell complexes as ideal learning representations. However, there is a lack of available machine learning methods suitable for learning on CW complexes. In this paper, we derive an explicit expression for the Wasserstein distance between cell complex signal distributions in terms of a Hodge-Laplacian matrix. This leads to a structurally meaningful measure to compare CW complexes and define the optimal transportation map. In order to simultaneously include both feature and structure information, we extend the Fused Gromov-Wasserstein distance to CW complexes. Finally, we introduce novel kernels over the space of probability measures on CW complexes based on the dual formulation of optimal transport.

Figures

Figures reproduced from arXiv: 2507.16569 by the authors.

**Figure 2.** In the figure above we can see the training losses, noise parameter, and test loss change for each [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** In the above figure we can see the two kernels visualized. On the left panel we see the exponential [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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write newline

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Gaussian Processes on Cellular Complexes

Mathieu Alain, So Takao, Brooks Paige, and Marc Peter Deisenroth. Gaussian processes on cellular complexes. arXiv preprint arXiv:2311.01198, 2023

work page Pith review arXiv 2023

[2] [2]

Graph classification gaussian processes via hodgelet spectral features, 2025

Mathieu Alain, So Takao, Xiaowen Dong, Bastian Rieck, and Emmanuel Noutahi. Graph classification gaussian processes via hodgelet spectral features, 2025. URL https://arxiv.org/abs/2410.10546

arXiv 2025

[3] [3]

Gaussian processes on distributions based on regularized optimal transport

Fran c ois Bachoc, Louis B \'e thune, Alberto Gonzalez-Sanz, and Jean-Michel Loubes. Gaussian processes on distributions based on regularized optimal transport. In International Conference on Artificial Intelligence and Statistics, pp.\ 4986--5010. PMLR, 2023

work page

[4] [4]

Identifying nonlinear dynamics with high confidence from sparse data

Bogdan Batko, Marcio Gameiro, Ying Hung, William Kalies, Konstantin Mischaikow, and Ewerton Vieira. Identifying nonlinear dynamics with high confidence from sparse data. SIAM Journal on Applied Dynamical Systems, 23 0 (1): 0 383--409, 2024. doi:10.1137/23M1560252. URL https://doi.org/10.1137/23M1560252

work page doi:10.1137/23m1560252 2024

[5] [5]

Mat \'e rn gaussian processes on graphs

Viacheslav Borovitskiy, Iskander Azangulov, Alexander Terenin, Peter Mostowsky, Marc Deisenroth, and Nicolas Durrande. Mat \'e rn gaussian processes on graphs. In International Conference on Artificial Intelligence and Statistics, pp.\ 2593--2601. PMLR, 2021

work page 2021

[6] [6]

Wasserstein exponential kernels

Henri De Plaen, Micha \"e l Fanuel, and Johan AK Suykens. Wasserstein exponential kernels. In 2020 International Joint Conference on Neural Networks (IJCNN), pp.\ 1--6. IEEE, 2020

work page 2020

[7] [7]

Stochastic optimization for large-scale optimal transport

Aude Genevay, Marco Cuturi, Gabriel Peyr\' e , and Francis Bach. Stochastic optimization for large-scale optimal transport. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016. URL https://proceedings.neurips.cc/paper_files/paper/2016/file/2a27b814...

work page 2016

[8] [8]

Menger’s theorem for topological spaces

William J Gilbert. Menger’s theorem for topological spaces. Fundamenta mathematicae, 75 0 (3): 0 291--295, 1972

work page 1972