Hessian Manifolds for Robust Graph Embeddings

Sim et al. embed directed graphs in pseudo-Riemannian manifolds but ignore curvature’s role in asymmetry. Liu’s work on Hessian manifolds shows how affine connections and Hessian metrics encode geometric constraints. This idea models directed graphs as Hessian manifolds: Edge directionality becomes a "torsion-free connection," while node relationships are encoded via Hessian potentials. The resulting embeddings satisfy Liu’s topological constraints (e.g., vanishing Euler characteristic), ensuring robustness to structural noise. Unlike Sim’s Minkowski embeddings, this approach respects the graph’s intrinsic affine geometry. Impact: New invariants for directed graphs and improved link prediction in asymmetric systems (e.g., citation networks).

References:

1. On Geometry and Topology of Hessian Manifolds. Hanwen Liu (2025).
2. Directed Graph Embeddings in Pseudo-Riemannian Manifolds. Aaron Sim, Maciej Wiatrak, Angus Brayne, Páidí Creed, S. Paliwal (2021). International Conference on Machine Learning.