Topological Neural Networks go Persistent, Equivariant, and Continuous
This work addresses the need for richer representations in machine learning for domains like molecular science, though it appears incremental as it unifies and extends existing methods.
The paper tackles the problem of enhancing neural networks by integrating topological neural networks (TNNs) with persistent homology (PH) to capture higher-order relational information beyond pairwise interactions, resulting in a framework called TopNets that achieves strong performance in tasks like antibody design and drug property prediction.
Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.