Characterize when curvature-adaptive alignment is necessary

Characterize which graphs and teacher kernels require curvature-adaptive treatment rather than uniform alignment for closing the spectral mismatch between GNN teachers and graph-free MLP students.

Background

The theoretical results establish upper-bound and locality properties: the GNN–MLP kernel gap is confined to message-passing receptive fields, and Ollivier–Ricci curvature acts as a local proxy for where the mismatch may concentrate. However, the paper does not establish matching lower bounds or prove that the identified curvature-stratified regimes are necessary for eliminating spectral mismatch.

The unresolved problem is therefore to provide a complete theoretical characterization of the graph structures and teacher kernels for which curvature-dependent supervision is genuinely required instead of uniform representation alignment.

References

We do not establish matching lower bounds, and in particular do not prove that the curvature-stratified regimes (Section~\ref{sec:theory-curvature}) are necessary for closing the spectral mismatch --- only that they are theoretically well-motivated and empirically effective. A complete characterization of which graphs and which teacher kernels demand curvature-adaptive treatment (rather than uniform alignment) is an open theoretical question.

— Distilling Graph Geometry: Knowledge Gap from GNNs to MLPs  (2610.10520 - Chen et al., 7 Oct 2026) in Appendix, Section "Limitations and Future Directions," paragraph "Upper-bound nature of the theoretical results"