# Topological Deep Learning: Going Beyond Graph Data

arXiv: https://arxiv.org/abs/2206.00606 · HTML: https://mlsanity.com/paper/2206.00606/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2022.

## Built on by (most cited first, 30 of 97)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networks](https://mlsanity.com/paper/2304.10031/) (2023)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [TopoBench: A Framework for Benchmarking Topological Deep Learning](https://mlsanity.com/paper/2406.06642/) (2024)
- [TopoX: A Suite of Python Packages for Machine Learning on Topological Domains](https://mlsanity.com/paper/2402.02441/) (2024)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Module](https://mlsanity.com/paper/2305.16174/) (2023)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks](https://mlsanity.com/paper/2410.06530/) (2024)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphs](https://mlsanity.com/paper/2312.09504/) (2023)
- [CIN++: Enhancing Topological Message Passing](https://mlsanity.com/paper/2306.03561/) (2023)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [Copresheaf Topological Neural Networks: A Generalized Deep Learning Framework](https://mlsanity.com/paper/2505.21251/) (2025)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Heterogeneous Sheaf Neural Networks](https://mlsanity.com/paper/2409.08036/) (2024)
- [On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theory](https://mlsanity.com/paper/2412.11521/) (2024)
- [HOG-Diff: Higher-Order Guided Diffusion for Graph Generation](https://mlsanity.com/paper/2502.04308/) (2025)
- [ICML 2023 Topological Deep Learning Challenge : Design and Results](https://mlsanity.com/paper/2309.15188/) (2023)
- [A Topological Deep Learning Framework for Neural Spike Decoding](https://mlsanity.com/paper/2212.05037/) (2022)
- [Hypergraph-MLP: Learning on Hypergraphs without Message Passing](https://mlsanity.com/paper/2312.09778/) (2023)
- [Higher-Order Topological Directionality and Directed Simplicial Neural Networks](https://mlsanity.com/paper/2409.08389/) (2024)
- [Integrating Physics and Topology in Neural Networks for Learning Rigid Body Dynamics](https://mlsanity.com/paper/2411.11467/) (2024)
- [HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations](https://mlsanity.com/paper/2505.15405/) (2025)
- [Nonlinear Sheaf Diffusion in Graph Neural Networks](https://mlsanity.com/paper/2403.00337/) (2024)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Manifold Topological Deep Learning for Biomedical Data](https://mlsanity.com/paper/2503.00175/) (2025)

## Abstract (arXiv)

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications.