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This research addresses a fundamental geometric mismatch in standard Graph Neural Networks by introducing Pseudo-Riemannian manifolds to handle heterophilic graphs where edges represent dissimilarity. The proposed PRAT model utilizes a learnable signed metric initialized via a new Graph Signature Index, demonstrating superior performance across fifteen benchmarks compared to existing heterophily-adapted architectures. This work advances the theoretical foundations of graph learning by relaxing positive-definite constraints to better align with real-world structural repulsion patterns.
Learning Pseudo-Riemannian Manifolds for Heterophilic Graphs via Graph Signature
arXiv:2611.06265v1 Announce Type: cross Abstract: Several widely used families of Graph Neural Networks carry a positive-definite geometric prior. Graph convolution read as heat diffusion needs a positive semi-definite operator, hyperbolic models place features on positive-definite Riemannian manifolds, and dot-product and GAT-style attention scores use positive-definite inner products. On heterophilic graphs, where edges often signify dissimilarity or structural repulsion, that prior is a mismatch: no positive-definite form can assign a negative squared distance to a direction the labels separate. To measure the mismatch, we introduce the Graph Signature Index ($s_{\mathcal{G}}$), the inertia of a class-compatibility matrix built from label topology alone, and we prove that no value of the edge or node homophily index in $(0,1)$ rules out a negative eigenvalue. We then propose the Pseudo-Riemannian Attention Network (PRAT), whose attention logit is a signed squared pseudo-distance under a learnable diagonal metric initialized from $s_{\mathcal{G}}$. The design is diagnose-then-learn: the signature supplies a sign structure that unconstrained training keeps outside near-zero directions, and the projection learns the rest of the metric around it. Across fifteen node-classification benchmarks PRAT outperforms or matches the strongest heterophily-adapted models. A controlled ablation ties that performance to the sign structure itself: with architecture and protocol fixed, forbidding the metric to be indefinite costs accuracy on the graphs $s_{\mathcal{G}}$ marks as indefinite and leaves the others unchanged.
Abstract:Several widely used families of Graph Neural Networks carry a positive-definite geometric prior. Graph convolution read as heat diffusion needs a positive semi-definite operator, hyperbolic models place features on positive-definite Riemannian manifolds, and dot-product and GAT-style attention scores use positive-definite inner products. On heterophilic graphs, where edges often signify dissimilarity or structural repulsion, that prior is a mismatch: no positive-definite form can assign a negative squared distance to a direction the labels separate. To measure the mismatch, we introduce the Graph Signature Index ($s_{\mathcal{G}}$), the inertia of a class-compatibility matrix built from label topology alone, and we prove that no value of the edge or node homophily index in $(0,1)$ rules out a negative eigenvalue. We then propose the Pseudo-Riemannian Attention Network (PRAT), whose attention logit is a signed squared pseudo-distance under a learnable diagonal metric initialized from $s_{\mathcal{G}}$. The design is diagnose-then-learn: the signature supplies a sign structure that unconstrained training keeps outside near-zero directions, and the projection learns the rest of the metric around it. Across fifteen node-classification benchmarks PRAT outperforms or matches the strongest heterophily-adapted models. A controlled ablation ties that performance to the sign structure itself: with architecture and protocol fixed, forbidding the metric to be indefinite costs accuracy on the graphs $s_{\mathcal{G}}$ marks as indefinite and leaves the others unchanged.
Accepted at NeurIPS 2026. 41 pages, including appendix
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2611.06265 [cs.LG]
(or arXiv:2611.06265v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2611.06265 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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