TopoU-Net: a U-Net architecture for topological domains

2026-05-11

AI Summary

Traditional neural networks struggle to model higher-order topological structures such as nodes, edges, faces, and hyperedges, often losing critical information when simplifying data into graphs or sequences. This work proposes a general U-Net architecture grounded in combinatorial complexes, replacing conventional spatial scales with “rank” as the hierarchical dimension. Cross-scale feature propagation is achieved through cells, incidence maps, and rank-wise pathways, while a bottleneck support ratio is introduced to quantify compression severity. The framework enables cohomological lifting and rank-matched skip connections across diverse topological domains, revealing the structural role of skip connections under high compression. Empirical results demonstrate that the model achieves state-of-the-art average accuracy on six out of eight node classification datasets and four out of five hypergraph benchmarks, with particularly pronounced gains on heterophilic graphs.

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.

Problem

• topological domains
• higher-order structure
• encoder-decoder architecture
• combinatorial complexes
• structured data

Innovation

• TopoU-Net
• combinatorial complexes
• rank-path
• bottleneck support ratio
• higher-order structure