Enhancing Robustness of Constant Curvature Graph Convolutional Network with Lipschitz Regularization

Abstract

Non-Euclidean spaces inherently enable high-fidelity embeddings for hierarchical and cyclical data due to their geometric properties. Existing approaches unify hyperbolic and spherical embeddings within the framework of constant curvature spaces. However, current methods for Lipschitz regularization remain limited to non-positive curvature geometries, such as hyperbolic and Euclidean spaces, and cannot be naturally extended to the general constant curvature setting. In this article, we present a rigorous Lipschitz analysis for constant curvature graph convolutional networks ( \(\kappa\) -GCNs) and enhance their robustness through Lipschitz regularization. We derive upper bounds for the Lipschitz constants across constant curvature spaces, thereby standardizing the Lipschitz limits of the \(\kappa\) -stereographic model. Furthermore, we incorporate these bounds into a regularization framework for \(\kappa\) -GCNs to improve stability and robustness. Experimental results demonstrate that the proposed regularization method often strengthens the robustness of \(\kappa\) -GCNs across various curvature regimes, particularly under Gaussian feature noise.