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

arXiv: https://arxiv.org/abs/2605.10091 · HTML: https://mlsanity.com/paper/2605.10091/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 37 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 28 of 28)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [UNet++: A Nested U-Net Architecture for Medical Image Segmentation](https://mlsanity.com/paper/1807.10165/) (2018)
- [Attention U-Net: Learning Where to Look for the Pancreas](https://mlsanity.com/paper/1804.03999/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Road Extraction by Deep Residual U-Net](https://mlsanity.com/paper/1711.10684/) (2017)
- [UNet 3+: A Full-Scale Connected UNet for Medical Image Segmentation](https://mlsanity.com/paper/2004.08790/) (2020)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [HNHN: Hypergraph Networks with Hyperedge Neurons](https://mlsanity.com/paper/2006.12278/) (2020)
- [Towards Sparse Hierarchical Graph Classifiers](https://mlsanity.com/paper/1811.01287/) (2018)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [HyperSAGE: Generalizing Inductive Representation Learning on Hypergraphs](https://mlsanity.com/paper/2010.04558/) (2020)
- [HodgeNet: Graph Neural Networks for Edge Data](https://mlsanity.com/paper/1912.02354/) (2019)
- [Cell Attention Networks](https://mlsanity.com/paper/2209.08179/) (2022)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [TopoX: A Suite of Python Packages for Machine Learning on Topological Domains](https://mlsanity.com/paper/2402.02441/) (2024)
- [Simplicial Attention Neural Networks](https://mlsanity.com/paper/2203.07485/) (2022)
- [Pooling Strategies for Simplicial Convolutional Networks](https://mlsanity.com/paper/2210.05490/) (2022)

## Abstract (arXiv)

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.