{"id":"c37d7df1-9484-407c-9d96-7f8e2b095653","arxiv_id":"2507.16569","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper defines Wasserstein and Fused Gromov-Wasserstein distances for CW complexes by substituting the Hodge Laplacian into existing graph OT formulas.","lead":"This paper applies optimal transport distances and kernels, originally built for graphs, to cell complexes by swapping the graph Laplacian for the Hodge Laplacian. The authors fit Gaussian processes with these kernels and report a small experiment on synthetic cell complexes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. 14 states W2 equals a trace expression that is in fact W2^2, contradicting Definition 1.5; the central closed-form distance is misstated.","rationale":"The reader's weakest assumption concerns the Gaussian modeling choice. I find a more immediate problem: the central identity in Eq. 14 is internally inconsistent with the paper's own definition of W2. The right-hand side is the squared Wasserstein-2 distance, not the distance. This alone invalidates the headline result as stated, although the fix (adding a square root) is simple. The Gaussian modeling concern remains valid but secondary. Because the experimental validation uses the misdefined distance, the empirical claims do not rescue the paper. The verdict stays REJECT, so the reader's verdict is unchanged.","tokens_in":10477,"tokens_out":6758,"duration_ms":67250,"concrete_test":"Take Sigma1 = diag(2,1) and Sigma2 = diag(3,2) in R^2. Compute the right-hand side of Eq. 14: Tr(Sigma1 + Sigma2) = 8, Tr(sqrt(Sigma1^{1/2} Sigma2 Sigma1^{1/2})) is approximately 3.864, so the RHS is approximately 0.273. The true W2 from Definition 1.5 / Eq. 12 is sqrt(0.273) which is approximately 0.522. Verify numerically with a Sinkhorn solver or the closed Gaussian formula; the two values will not coincide. Repeating this check for a handful of random full-rank covariance pairs settles whether the missing square root is systematic.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is the closed form for W2 between CW-complex signal Gaussians. Definition 1.5 (Eq. 3) defines Wp as the p-th root of the transport cost, so for p=2 the Wasserstein distance is the square root of the integral in Eq. 13. The standard Gaussian formula (Dowson–Landau/Givens) is W2^2 = Tr(Sigma1 + Sigma2) - 2 Tr(sqrt(Sigma1^{1/2} Sigma2 Sigma1^{1/2})). Eq. 14 reproduces the right-hand side of that formula and assigns it to W2 without the outer square root; Eq. 13 also writes W2 as the integral itself. Thus the equation is internally inconsistent with the paper's own Definition 1.5. Since the kernels in Eqs. 18-19 and all reported losses (Table 1) are evaluated using distances from Eq. 14, the experimental numbers do not correspond to a Wasserstein-2 distance. The pseudoinverse placement in the trace is also untested for singular covariances (Hodge Laplacians always have nontrivial kernels), so the formula needs an argument for degenerate Gaussians.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10779,"tokens_out":10783,"duration_ms":106459,"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":[{"comment":"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.","section":"§3.3, Eqs. (13)–(14)"},{"comment":"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.","section":"§3.3, Definition 3.4"},{"comment":"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.","section":"§3.1, Theorem 3.1"},{"comment":"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.","section":"§3.4, Definition 3.5"},{"comment":"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.","section":"§4.3, Definition 4.1"},{"comment":"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.","section":"§6 and Table 1"}],"minor_comments":[{"comment":"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.","section":"§4.1, Eq. (18)"},{"comment":"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.","section":"§3.4, Eq. (16)"},{"comment":"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).","section":"Algorithm 1"},{"comment":"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.","section":"Definition 3.4"},{"comment":"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.","section":"Theorem 3.1 proof"}],"recommendation":"major_revision","confidential_remarks":"The core formula in Eq. (14) is a known Gaussian optimal-transport result applied to a new family of covariance matrices; the novelty lies in the adaptation to cell complexes and in the proposed kernels. The square-root error and the ill-posed FGW definition are fixable in principle, but the revision will be substantial. I would also ask the editor to weigh the absence of baselines and the unvalidated Gaussian modeling assumption when considering resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nBeyond 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.\n\nThe 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.\n\nThe 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.\n\nNet: 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.","headline":"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.","tokens_in":11260,"tokens_out":5078,"would_cite":false,"duration_ms":51574,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["optimal transport","Wasserstein distance","CW complexes","Hodge Laplacian","Gaussian processes","Fused Gromov-Wasserstein","kernel methods","cell complexes"],"falsifier":"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.","tokens_in":10304,"feed_emoji":"📐","tokens_out":11263,"duration_ms":101993,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hodge Laplacians give cell complexes a Wasserstein distance","feed_subtitle":"A single trace formula yields a structural distance, an optimal transport map, and kernels for learning on CW complexes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Gives the CW-complex construction and the homotopy-equivalence-with-a-simplicial-complex theorem used in Theorem 3.1.","marker":"Hatcher (2002)"},{"why":"Supplies the graph signal-distribution model, the Gaussian Wasserstein identity, and the optimal transport map that Definition 3.4 extends to CW complexes.","marker":"Petric Maretic et al. (2019)"},{"why":"Introduces the Fused Gromov-Wasserstein distance whose feature-plus-structure formulation Definition 3.5 adapts.","marker":"Vayer et al. (2019)"},{"why":"Provides the weighted Hodge Laplacian matrix representation with cell weights and boundary matrices used in Definitions A.5, A.6, and 3.5.","marker":"Alain et al. (2023)"},{"why":"Provides the linear embedding result for finite simplicial complexes used in the proof of Theorem 3.1.","marker":"Lazarus (2020)"},{"why":"Supplies the fact that a countable product of Polish spaces is Polish, used to define a common space for the two measures.","marker":"Kechris (2012)"},{"why":"Supplies the Whitney embedding theorem used to turn a regular neighborhood of the complex into a Euclidean embedded Polish space.","marker":"Lee (2012)"},{"why":"Gives the bandwidth threshold guaranteeing positive semidefiniteness of exponential distance kernels and the spectral-truncation feature map used in Section 4.3.","marker":"De Plaen et al. (2020)"},{"why":"Supplies the foundational optimal-transport and Wasserstein-distance definitions the paper builds on.","marker":"Villani (2021)"}],"fun_headline_variants":["Wasserstein distance from Hodge Laplacians for cell complexes","Cell complex kernels via optimal transport and Hodge Laplacians","Closed-form Wasserstein for CW complexes","New kernels link cell complexes with Wasserstein geometry","Hodge Laplacians unlock optimal transport on cell complexes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Wasserstein distance from Hodge Laplacians for cell complexes","Cell complex kernels via optimal transport and Hodge Laplacians","Closed-form Wasserstein for CW complexes","New kernels link cell complexes with Wasserstein geometry","Hodge Laplacians unlock optimal transport on cell complexes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000199,"raw_usage":{"total_tokens":1371,"prompt_tokens":942,"completion_tokens":429,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":558,"completion_tokens_details":{"reasoning_tokens":351}},"tokens_in":558,"tokens_out":429,"duration_ms":4269,"temperature":1.0,"reasoning_tokens":351,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:06:20.046469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Algebraic Topology","cited_arxiv_id":null,"evidence_quote":"Gives the CW-complex construction and the homotopy-equivalence-with-a-simplicial-complex theorem used in Theorem 3.1."},{"cited_title":"Got: An optimal transport framework for graph comparison","cited_arxiv_id":null,"evidence_quote":"Supplies the graph signal-distribution model, the Gaussian Wasserstein identity, and the optimal transport map that Definition 3.4 extends to CW complexes."},{"cited_title":"Optimal transport for structured data with application on graphs","cited_arxiv_id":null,"evidence_quote":"Introduces the Fused Gromov-Wasserstein distance whose feature-plus-structure formulation Definition 3.5 adapts."},{"cited_title":"Gaussian Processes on Cellular Complexes","cited_arxiv_id":"2311.01198","evidence_quote":"Provides the weighted Hodge Laplacian matrix representation with cell weights and boundary matrices used in Definitions A.5, A.6, and 3.5."},{"cited_title":"Embedding in euclidean spaces: the double dimension case","cited_arxiv_id":null,"evidence_quote":"Provides the linear embedding result for finite simplicial complexes used in the proof of Theorem 3.1."},{"cited_title":"Classical descriptive set theory, volume 156","cited_arxiv_id":null,"evidence_quote":"Supplies the fact that a countable product of Polish spaces is Polish, used to define a common space for the two measures."},{"cited_title":"Smooth manifolds","cited_arxiv_id":null,"evidence_quote":"Supplies the Whitney embedding theorem used to turn a regular neighborhood of the complex into a Euclidean embedded Polish space."},{"cited_title":"Wasserstein exponential kernels","cited_arxiv_id":null,"evidence_quote":"Gives the bandwidth threshold guaranteeing positive semidefiniteness of exponential distance kernels and the spectral-truncation feature map used in Section 4.3."},{"cited_title":"Topics in optimal transportation, volume 58","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational optimal-transport and Wasserstein-distance definitions the paper builds on."}],"review_version":1}