Title: Heat Kernel Goes Topological
Primary area: learning on graphs and other geometries & topologies
Keywords: Topological Deep Learning, Heat Kernel Signature, Spectral Methods

Abstract: We propose a scalable framework for topological deep learning that uses the Heat Kernel Signature (HKS) as a node descriptor on combinatorial complexes. Unlike existing topological neural networks, which rely on expensive higher-order message passing, our approach leverages the Laplacian operator to compute multiscale HKS embeddings that are permutation-equivariant and readily integrated into traditional general deep learning methods. This enables efficient learning while preserving rich structural information.
Theoretically, our method achieves maximal expressivity within the limits of spectral techniques, distinguishing complexes up to isospectral equivalence. Empirically, it outperforms existing topological baselines in computational efficiency while maintaining competitive accuracy on molecular property prediction benchmarks. On specialised topological tasks, it shows superior ability to separate complex structures and overcome known blind spots of prior methods.
Overall, our results establish HKS-based node features as a powerful primitive for topological representation learning, offering a principled trade-off between expressivity and scalability and opening new directions for applying topological methods in molecular and geometric machine learning.

TL;DR: We define novel heat kernel descriptors for combinatorial complexes, outperforming existing topological methods in terms of computational efficiency.