# Topological Neural Tangent Kernel

arXiv: https://arxiv.org/abs/2605.01110 · HTML: https://mlsanity.com/paper/2605.01110/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 19 references that are in the map.
- First posted on arXiv 1 May 2026.

## Builds on (most cited first, 15 of 15)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Graph Neural Tangent Kernel: Fusing Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/1905.13192/) (2019)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Simplicial Closure and higher-order link prediction](https://mlsanity.com/paper/1802.06916/) (2018)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Graph Neural Tangent Kernel: Convergence on Large Graphs](https://mlsanity.com/paper/2301.10808/) (2023)
- [Spectral Detection of Simplicial Communities via Hodge Laplacians](https://mlsanity.com/paper/2108.06547/) (2021)
- [The collective vs individual nature of mountaineering: a network and simplicial approach](https://mlsanity.com/paper/2201.13344/) (2022)

## Abstract (arXiv)

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.