Volume-Weighted Hodge Convolution
Implementation & benchmark of arXiv:2607.18692 — Geometry-Induced Hodge Stars on Rips and Dowker--Rips Complexes
Source paper: Geometry-Induced Hodge Stars on Rips and Dowker--Rips Complexes arXiv:2607.18692 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace the ordinary combinatorial Hodge propagation in a simplicial neural network with a geometry-induced weighted Hodge Laplacian built from Euclidean simplex volumes. The operator preserves the harmonic/topological subspace while changing the positive spectrum according to the shape and scale of the simplices, allowing message passing to distinguish geometrically meaningful cells that have identical incidence patterns.
Formulas
Mathematical statement
For an oriented simplicial complex K, let B_k be the signed boundary matrix mapping k-simplices to (k-1)-simplices, and let W_k=diag{w_k(σ):σ∈K_k} be a positive diagonal matrix assigning weight w_k(σ)>0 to each k-simplex σ. The paper's weighted Hodge Laplacian on k-cochains is $$L_k^W=W_k^{-1}B_k^T W_{k-1}B_k+B_{k+1}W_{k+1}^{-1}B_{k+1}^T W_k.$$ The first term is the weighted down-Laplacian and the second is the weighted up-Laplacian; omit a term when the relevant boundary matrix does not exist. The weighted inner product is ⟨u,v⟩_{W_k}=u^T W_kv, so L_k^W is self-adjoint in this inner product. For every choice of strictly positive weights, ker(L_k^W) has dimension β_k, the kth Betti number, while the nonzero eigenvalues change with the weights. For the volume star, define w_k(σ)=Vol_k(σ), where Vol_k is the Euclidean k-dimensional simplex volume; in coordinates σ=(x_0,…,x_k), $$Vol_k(σ)=\frac{1}{k!}\sqrt{\det\left((x_i-x_0)^T(x_j-x_0)\right)_{i,j=1}^{k}}.$$ For numerical stability use w_k(σ)=max(Vol_k(σ),ε_w), optionally normalized by the median weight at each dimension. A neural layer can use a scaled polynomial filter of the adapted operator, for example $$H^{(\ell+1)}=\phi\left(\sum_{r=0}^{R}\Theta_r(\widetilde L_k^W)^r H^{(\ell)}\right),\qquad \widetilde L_k^W=\frac{2L_k^W}{\lambda_{\max}}-I,$$ where H^(ℓ) has one row per k-simplex, Θ_r are learnable channel-mixing matrices, φ is an activation, and λ_max is an estimated largest eigenvalue.
Implementation notes
Integrate this at the simplicial message-passing layer for one chosen cochain degree k, initially k=1 for edge features or k=2 for face features. Construct a Rips complex from point coordinates X using the same radius and maximum simplex dimension as the baseline model. Orient every simplex once and build sparse signed incidence matrices B_k. For each k-simplex, compute its Gram matrix G_ij=(x_i-x_0)^T(x_j-x_0) and set w_k=sqrt(max(det(G),0))/k!; clamp to ε_w, then divide all weights of a fixed dimension by their median. Form sparse W_k and evaluate the two terms in the paper's formula without materializing dense inverses: multiply by reciprocal diagonal vectors W_k^{-1}. Given feature matrix H with one row per k-simplex, compute Y_down=W_k^{-1} B_k^T(W_{k-1}(B_kH)) and Y_up=B_{k+1}(W_{k+1}^{-1}(B_{k+1}^T(W_kH))); set L_H=Y_down+Y_up. Use either H_next=MLP([H,L_H]) or a degree-2 filter H_next=φ(Θ_0H+Θ_1L_H+Θ_2L_H(L_H)). Estimate λ_max by 10-step power iteration on L_k^W and use the rescaled operator only if polynomial filtering is used. The mathematical quantities are the incidence matrices, simplex volumes, weighted Laplacian, and its guaranteed Betti-preserving kernel; empirically estimate λ_max and tune ε_w, filter order, and normalization. First test on a small point-cloud classification benchmark such as ModelNet40 subsampled to 256 points, comparing an unweighted Hodge layer, a graph-only layer, and the volume-weighted layer at equal parameter count and FLOPs. Measure accuracy, calibration, training loss variance, gradient norms, and sensitivity to anisotropic coordinate transformations. A useful success signal is a consistent accuracy gain on geometrically distorted or nonuniformly sampled clouds, with no increase in NaNs or optimization instability; a stronger result is improved accuracy at equal number of simplices and a smaller performance drop when the Rips radius changes.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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