Topological Neural Tangent Kernel
Graph neural networks miss higher-order topology—TopoNTK fills the gap with Hodge theory.
Graph neural tangent kernels (NTKs) have long provided a principled infinite-width theory for graph neural networks, but they are fundamentally limited to pairwise interactions. Many real-world relational systems—such as social networks or biological pathways—involve higher-order (simplicial) relationships. In a new paper, Sanjukta Krishnagopal from (presumably) an academic institution introduces the Topological Neural Tangent Kernel (TopoNTK), which extends NTK theory to simplicial complexes. By combining lower Hodge interactions (coupling through shared vertices) with upper Hodge interactions (coupling through filled simplices), TopoNTK becomes sensitive to topological structure that graph kernels cannot see—for example, two complexes with the same graph but different filled triangles will yield distinct kernels.
The kernel also brings interpretability: edge signals decompose into gradient, harmonic, and circulation components via Hodge theory, and the kernel's spectrum reveals a topological spectral bias. Modes with large eigenvalues are learned quickly, while global harmonic modes (which persist through a residual channel) have smaller eigenvalues and are learned more slowly. The paper proves expressivity, Hodge-alignment, spectral learning, and stability properties, and validates them on synthetic simplicial tasks and DBLP higher-order link prediction. The results suggest that topology provides a coordinate system that makes relational learning more faithful, interpretable, and effective.
- TopoNTK combines lower and upper Hodge interactions to capture topology invisible to standard graph kernels.
- Hodge decomposition yields interpretable learning geometry: gradient, harmonic, and local circulation components.
- Experiments on DBLP link prediction show topology improves relational learning performance and interpretability.
Why It Matters
TopoNTK makes graph models topology-aware, boosting accuracy and interpretability for complex relational data like social networks and biology.