Heat Kernel Goes Topological
This work advances topological deep learning by providing scalable representations for molecular classification and property prediction, though it is incremental as it builds on existing topological frameworks.
The paper tackles the computational expense of higher-order message passing in topological neural networks by introducing a Laplacian operator on combinatorial complexes to compute efficient heat kernels as node descriptors. The method achieves competitive performance on molecular datasets and significantly outperforms existing topological methods in computational efficiency.
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.