Topological Scattering over Product Cell Complexes

In this paper, we propose a non-parametric task-agnostic representation learning method for cell complexes (CCs). Specifically, we propose a scattering transform for CCs that extends geometric scattering to CCs. In addition, we introduce scattering transforms for product cell complexes (PCCs), which are a particular type of CCs that can be factored as the Cartesian product of two smaller simplicial complexes (SCs). We utilize the intrinsic product structure of PCCs to develop separable filters across the two constituent factors of PCCs. The proposed scattering transform for PCCs is computationally efficient, as it can be expressed in terms of scattering on the individual SCs. The effectiveness of the proposed model is demonstrated through experiments on various tasks, such as trajectory classification, missing edge feature estimation, and edge feature prediction, on both synthetic and real-world PCCs.