Topological Neural Networks go Persistent, Equivariant and Continuous

Topological Neural Networks (TNNs) have enabled representations using higher dimensional simplicial complexes. Concurrently, persistence homology methods have undergone rapid strides, offering rich topological descriptors that improve the expressivity of GNNs. However, the integration of these methods to increase the expressivity of TNNs, and adaptation in handling geometric complexes, remains an unexplored frontier. We introduce TopNets, extending the concept of TNNs by unifying them with persistent homology (PH), equivariance and making them continuous. This framework provides a generalized approach that encompasses various methods at the intersection of PH and TNNs. TopNets enhances the expressiveness of Equivariant Message Passing (MP) simplicial networks, allowing them to acquire high-dimensional simplex features alongside topological embeddings generated through geometric color filtrations in an $\mathrm{E}(n)$-equivariant manner. Empirical evaluation demonstrates the efficacy of the proposed method across diverse tasks such as graph classification, drug property prediction, and generative design.