Source
Bodnar et al. (2022), LIT-1811, §5 (pp. 7–8), §6 (pp. 8–9) and Appendix F (pp. 27–29) of arXiv 2202.04579v4; read in NOTE-1332.
What was actually shown
The model. Each restriction map is F_{v⊴e} = σ(V[x_v‖x_u]), recomputed at every layer from the current features, in diagonal, orthogonal (Householder) or general form, inside a residual discretisation of sheaf diffusion (eqs. 5, 6; p. 8). Stalk dimension is 1 to 5 (p. 8).
Synthetic. On a connected balanced bipartite graph with overlapping Gaussian features, a d = 1 diffusion with learned general maps separates the classes as diffusion time grows, while learned symmetric maps do not (Fig. 5, p. 8), as Prop. 9 predicts. The histogram of learned transports is negative for all edges, as Prop. 10 predicts (p. 9). With three classes and d = 2 the learned rotations peak near 2 radians on heterophilic edges, close to 2π/3, but neither the d = 1 nor the d = 2 model reaches the ideal sheaf’s performance (Fig. 7, p. 27).
Benchmarks. On nine node-classification datasets, each over the same ten splits, an NSD variant is first on Texas, Wisconsin, Film, Squirrel and Cornell, second on Chameleon, and within about 1% of the best on Citeseer, Pubmed and Cora (Table 1, p. 9). Continuous-time versions do less well than the discrete ones but beat other continuous models (Table 3, p. 28).
What could have failed. The learned maps in the synthetic case could have stayed positive, or the model could have trailed the heterophily- specific baselines on the benchmarks.
What this does not say
- That the learned maps on real graphs are class-aligned. No learned map is shown for any benchmark dataset. The accuracy is consistent with the theory’s mechanism and with others: the model is also a feature- conditioned message function, “a form of GNN-FiLM layer” (p. 10), and such models “have been recently shown to perform well empirically in heterophilic settings” (p. 10).
- That the margins are reliable. The heterophilic datasets have 183 to 7,600 nodes, standard deviations run to ±9 points, and the baselines are copied from other papers’ runs on the same splits (p. 9). Several first places are within one standard deviation of the second.
- That learning recovers a ground truth. In the synthetic case the target sheaf is known because the labels define it. On real data there is no such sheaf to compare with, so “learns the geometry” has no test here (QUESTION-130).
- Anything about larger graphs or other tasks. Only node classification on these nine sets is reported.
Note of 2026-10-30: promoted
Promoted as a report of what the paper measured (ADR-065). The paper’s explanation of its accuracy, that the learned maps have the class-aligned geometry its theory prescribes, is THEORY-1220, Proposed, which extends this entry and keeps its old condition.