Heat Kernel Goes Topological

TL;DR

This paper introduces a scalable topological framework using a Laplacian operator on combinatorial complexes to efficiently compute heat kernels, enabling expressive node descriptors for neural networks.

Contribution

It presents a novel Laplacian-based method for topological neural networks that is both computationally efficient and maximally expressive for complex structures.

Findings

Outperforms existing methods in computational efficiency

Achieves competitive results on molecular datasets

Excels at distinguishing complex topological structures

Abstract

Topological neural networks have emerged as powerful successors of graph neural networks. However, they typically involve higher-order message passing, which incurs significant computational expense. We circumvent this issue with a novel topological framework that introduces a Laplacian operator on combinatorial complexes (CCs), enabling efficient computation of heat kernels that serve as node descriptors. Our approach captures multiscale information and enables permutation-equivariant representations, allowing easy integration into modern transformer-based architectures. Theoretically, the proposed method is maximally expressive because it can distinguish arbitrary non-isomorphic CCs. Empirically, it significantly outperforms existing topological methods in terms of computational efficiency. Besides demonstrating competitive performance with the state-of-the-art descriptors on standard molecular datasets, it exhibits superior capability in distinguishing complex topological structures and avoiding blind spots on topological benchmarks. Overall, this work advances topological deep learning by providing expressive yet scalable representations, thereby opening up exciting avenues for molecular classification and property prediction tasks.