---
title: "Learning Pseudo-Riemannian Manifolds for Heterophilic Graphs via Graph Signature"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2611.06265v1"
permalink: "https://feed.astralor.com/items/itm_0c6f261d02bd23d61c363db7"
published: "2026-11-04T10:14:42.000Z"
score: 60
channel: "tech"
tags: ["Learning Pseudo-Riemannian Manifolds","Heterophilic Graphs","Graph Signature Several","Graph Neural Networks","Graph","Several"]
---

# Learning Pseudo-Riemannian Manifolds for Heterophilic Graphs via Graph Signature

## At a glance

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 posi…

## Key points

- 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-d…
- 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.

## Full text

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.

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