Topological Neural Tangent Kernel
For researchers working on graph neural networks and topological deep learning, this work addresses the limitation of graph models in handling higher-order interactions by providing a principled kernel that is sensitive to topology, though it is an incremental extension of neural tangent kernels to simplicial complexes.
The paper introduces the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing that captures higher-order interactions beyond pairwise graphs. It demonstrates that TopoNTK provides interpretable learning geometry and spectral bias, validated on synthetic tasks and DBLP link prediction.
Graph neural tangent kernels give a principled infinite-width theory for graph neural networks, but inherit a basic limitation of graph models: they see only pairwise structure. Many relational systems contain higher-order interactions that are more naturally represented by simplicial complexes. We introduce the Topological Neural Tangent Kernel (TopoNTK), an infinite-width kernel for simplicial message passing on edge features. TopoNTK combines lower Hodge interactions, capturing graph-like coupling through shared vertices, with upper Hodge interactions, capturing coupling through filled simplices. This makes the kernel sensitive to topology invisible to graph kernels, allowing complexes with the same graph but different filled simplices to induce different kernels. Beyond expressivity, the Hodge structure gives the kernel an interpretable learning geometry. Edge signals decompose into gradient-like, harmonic, and local circulation components, and the spectrum of the TopoNTK determines how quickly each component is learned. This yields a topological form of spectral bias: components aligned with large-eigenvalue modes are learned quickly, while global harmonic modes, retained through the residual channel, often lie at smaller eigenvalues and are learned more slowly. We prove expressivity, Hodge-alignment, spectral learning, and stability properties, and validate them on synthetic simplicial tasks and DBLP higher-order link prediction. The results show that topology is not merely extra structure; it can provide coordinates that make relational learning more faithful, interpretable, and effective.