Hourglass Persistence for Graphs, Simplices, and Cells

Persistent homology (PH) based schemes help encode information, such as cycles, and are thus increasingly being integrated with graph neural networks (GNNs) and higher order message-passing networks. Many PH based schemes in graph learning employ inclusion-based filtration mechanisms that trace a sequence of subgraphs of increasing size, maintaining bookkeeping information about the evolution (e.g., in terms of birth and death of components). We offer a novel perspective that goes beyond this inclusion paradigm. Specifically, we introduce topological descriptors for graphs, simplices, and cells that interleave a sequence of inclusions with a sequence of contractions and related families parametrized by two functions. The resulting descriptors on the extended sequence are provably more expressive than many existing PH methods with suitable stability conditions. Empirical results substantiate the merits of the proposed approach.