AI summaryⓘ
The authors introduce TopoU-Net, a new neural network design that works well on complex data made of points, edges, faces, and higher-order structures, rather than just grids or graphs. They rethink the U-Net architecture by using a hierarchy based on combinatorial complexes, which naturally organize data at different levels called ranks. This allows the network to move information up and down these ranks with skip connections linking matching levels, adapting to various data types like graphs, hypergraphs, and meshes. Their method improves accuracy on many classification tasks, especially when data connections are complex, and shows that some design choices like skip connections are crucial when compressing information too much. Overall, the authors offer a flexible encoder-decoder framework for data with rich higher-order structure.
U-Netcombinatorial complextopological domainincidence mapcochainencoder-decoderskip connectionhypergraphbottleneckheterophilic graph
Authors
Gaurav Gaurav, Ibrahem ALJabea, Yaroslav Zakomornyy, Eric Frank, Mohamed Elhamdadi, Theodore Papamarkou, Mustafa Hajij
Abstract
Many modern datasets mix points, edges, regions, groups, objects, events, hyperedges, and relations. Yet neural architectures often force such data into grids, graphs, or sequences, obscuring higher-order structure and making encoder-decoder designs domain-specific. We view U-Net not as a grid-specific architecture, but as a hierarchical encoder-decoder principle: representation spaces, transport maps between levels, and skip connections between matched levels. Combinatorial complexes naturally supply these ingredients through cells, incidences, and ranks. We introduce TopoU-Net, a rank-path U-Net for topological domains. Given a path from an input rank to a bottleneck rank and back, the encoder lifts cochains upward along incidence maps, the decoder transports them downward, and skip connections merge features at matched ranks. Rank replaces spatial scale: choosing paths through nodes, edges, faces, hyperedges, or global cells becomes the central architectural decision. A key quantity is the bottleneck support ratio, the number of cells at the bottleneck relative to the number of cells at the input rank. This ratio is fixed by the complex and chosen path rather than by arbitrary pooling, and it clarifies when skip connections are optional, useful, or structurally important. Across node classification, graph classification, hypergraph node classification, mesh classification, and image reconstruction, TopoU-Net provides a reusable encoder-decoder template for higher-order structured data. Among the evaluated baselines, it achieves the strongest mean accuracy on six of eight node-classification datasets and four of five hypergraph datasets, with the largest gains on heterophilic graphs. Ablations show that removing skip connections is most damaging under severe bottleneck compression.