# Dist2Cycle: A Simplicial Neural Network for Homology Localization

arXiv: https://arxiv.org/abs/2110.15182 · HTML: https://mlsanity.com/paper/2110.15182/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 49 references that are in the map.
- First posted on arXiv 28 Oct 2021.

## Builds on (most cited first, 24 of 24)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Hypergraph Neural Networks](https://mlsanity.com/paper/1809.09401/) (2018)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Handling Missing Data with Graph Representation Learning](https://mlsanity.com/paper/2010.16418/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [HodgeNet: Graph Neural Networks for Edge Data](https://mlsanity.com/paper/1912.02354/) (2019)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Persistence Homology of Networks: Methods and Applications](https://mlsanity.com/paper/1907.08708/) (2019)
- [Can neural networks learn persistent homology features?](https://mlsanity.com/paper/2011.14688/) (2020)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [A Notion of Harmonic Clustering in Simplicial Complexes](https://mlsanity.com/paper/1910.07247/) (2019)
- [Topological Deep Learning](https://mlsanity.com/paper/2101.05778/) (2021)
- [Efficient algorithms for computing a minimal homology basis](https://mlsanity.com/paper/1801.06759/) (2018)
- [Pseudoinverse Graph Convolutional Networks: Fast Filters Tailored for Large Eigengaps of Dense Graphs and Hypergraphs](https://mlsanity.com/paper/2008.00720/) (2020)
- [Minimum cycle and homology bases of surface embedded graphs](https://mlsanity.com/paper/1607.05112/) (2016)

## Built on by (most cited first, 16 of 16)

- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Simplicial Attention Networks](https://mlsanity.com/paper/2204.09455/) (2022)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Simplicial Representation Learning with Neural k-Forms](https://mlsanity.com/paper/2312.08515/) (2023)
- [Cycle Invariant Positional Encoding for Graph Representation Learning](https://mlsanity.com/paper/2311.14333/) (2023)
- [ICML 2023 Topological Deep Learning Challenge : Design and Results](https://mlsanity.com/paper/2309.15188/) (2023)
- [Topo-MLP : A Simplicial Network Without Message Passing](https://mlsanity.com/paper/2312.11862/) (2023)
- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [GEFL: Extended Filtration Learning for Graph Classification](https://mlsanity.com/paper/2406.02732/) (2024)
- [Binarized Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2405.04098/) (2024)
- [Continuous Simplicial Neural Networks](https://mlsanity.com/paper/2503.12919/) (2025)
- [TopoGeoScore: A Self-Supervised Source-Only Geometric Framework for OOD Checkpoint Selection](https://mlsanity.com/paper/2605.08870/) (2026)
- [Privacy-Preserving Representation Learning for Text-Attributed Networks with Simplicial Complexes](https://mlsanity.com/paper/2302.04383/) (2023)
- [NervePool: A Simplicial Pooling Layer](https://mlsanity.com/paper/2305.06315/) (2023)
- [Tumor-Centered Patching for Enhanced Medical Image Segmentation](https://mlsanity.com/paper/2308.12168/) (2023)
- [Learning From Simplicial Data Based on Random Walks and 1D Convolutions](https://mlsanity.com/paper/2404.03434/) (2024)

## Abstract (arXiv)

Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the k-homological features of simplicial complexes. By spectrally manipulating their combinatorial k-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each k-simplex from the nearest "optimal" k-th homology generator, effectively providing an alternative to homology localization.