mlsanityData to 14 Oct 2026

Convolutional Learning on Simplicial Complexes

arXiv 2301.11163 ↗First posted 26 Jan 2023cs.LGcited by 27

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Abstract (arXiv)

We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently and captures the inter-simplicial couplings, generalizing state-of-the-art. Upon studying symmetries of the simplicial domain and the data space, it is shown to be permutation and orientation equivariant, thus, incorporating such inductive biases. Based on the Hodge theory, we perform a spectral analysis to understand how SCCNNs regulate data in different frequencies, showing that the convolutions via faces and cofaces operate in two orthogonal data spaces. Lastly, we study the stability of SCCNNs to domain deformations and examine the effects of various factors. Empirical results show the benefits of higher-order convolutions and inter-simplicial couplings in simplex prediction and trajectory prediction.

Key facts. Cited by 27 papers in the map (data through 14 Oct 2026). Builds on 38 of its 78 references that are in the map. First posted on arXiv 26 Jan 2023.

Builds on

38 of 78 references.

References · 38

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,487
  3. 2019Simplifying Graph Convolutional Networkscited by 1,477
  4. 2016Variational Graph Auto-Encoderscited by 1,382
  5. 2018Relational inductive biases, deep learning, and graph networkscited by 1,360
  6. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,167
  7. 2018Link Prediction Based on Graph Neural Networkscited by 832
  8. 2019Human Motion Trajectory Prediction: A Surveycited by 232
  9. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 221
  10. 2021Weisfeiler and Lehman Go Cellular: CW Networkscited by 218
  11. 2019Stability Properties of Graph Neural Networkscited by 127
  12. 2018Construction of the Literature Graph in Semantic Scholarcited by 118
26 more
  1. 2020Simplicial Neural Networkscited by 110
  2. 2019Topological Signal Processing over Simplicial Complexescited by 96
  3. 2018Simplicial Closure and higher-order link predictioncited by 93
  4. 2018Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplaciancited by 86
  5. 2021Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyondcited by 82
  6. 2020Graph Neural Networks: Architectures, Stability and Transferabilitycited by 80
  7. 2021Principled Simplicial Neural Networks for Trajectory Predictioncited by 77
  8. 2020Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networkscited by 75
  9. 2013Discrete Signal Processing on Graphs: Frequency Analysiscited by 73
  10. 2020Cell Complex Neural Networkscited by 67
  11. 2021What are higher-order networks?cited by 66
  12. 2022Simplicial Attention Networkscited by 41
  13. 2021Simplicial Convolutional Neural Networkscited by 40
  14. 2019HodgeNet: Graph Neural Networks for Edge Datacited by 39
  15. 2018Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Spacecited by 37
  16. 2019Stability of Graph Scattering Transformscited by 34
  17. 2021Signal Processing on Cell Complexescited by 31
  18. 2021Topological Signal Processing over Cell Complexescited by 24
  19. 2021BScNets: Block Simplicial Complex Neural Networkscited by 21
  20. 2022Simplicial Convolutional Filterscited by 18
  21. 2021Simplicial Complex Representation Learningcited by 17
  22. 2021Dist2Cycle: A Simplicial Neural Network for Homology Localizationcited by 16
  23. 2021Finite Impulse Response Filters for Simplicial Complexescited by 15
  24. 2022Convolutional Filtering in Simplicial Complexescited by 8
  25. 2022Stability of Aggregation Graph Neural Networkscited by 7
  26. 2019Extrapolating paths with graph neural networkscited by 5

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 6; 2024: 10; 2025: 5; 2026: 6.

27 papers.

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  1. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  2. 2023Architectures of Topological Deep Learning: A Survey of Message-Passing Topological Neural Networkscited by 43
  3. 2023Generalized Simplicial Attention Neural Networkscited by 27
  4. 2024TopoBench: A Framework for Benchmarking Topological Deep Learningcited by 24
  5. 2024TopoX: A Suite of Python Packages for Machine Learning on Topological Domainscited by 22
  6. 2024E(n) Equivariant Topological Neural Networkscited by 22
  7. 2023From Latent Graph to Latent Topology Inference: Differentiable Cell Complex Modulecited by 19
  8. 2024TopoTune : A Framework for Generalized Combinatorial Complex Neural Networkscited by 18
  9. 2023Facilitating Graph Neural Networks with Random Walk on Simplicial Complexescited by 14
  10. 2024MANTRA: The Manifold Triangulations Assemblagecited by 12
  11. 2024On the Theoretical Expressive Power and the Design Space of Higher-Order Graph Transformerscited by 8
  12. 2023ICML 2023 Topological Deep Learning Challenge : Design and Resultscited by 7
15 more
  1. 2024Higher-Order Topological Directionality and Directed Simplicial Neural Networkscited by 5
  2. 2025HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representationscited by 5
  3. 2024Demystifying Higher-Order Graph Neural Networkscited by 4
  4. 2024Molecular topological deep learning for polymer property predictioncited by 2
  5. 2025Continuous Simplicial Neural Networkscited by 2
  6. 2025Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexescited by 2
  7. 2025Heat Kernel Goes Topologicalcited by 1
  8. 2023NervePool: A Simplicial Pooling Layercited by 0
  9. 2025Directed Semi-Simplicial Learning with Applications to Brain Activity Decodingcited by 0
  10. 2026From Classical to Topological Neural Networks Under Uncertaintycited by 0
  11. 2026Topology of Reasoning: Retrieved Cell Complex-Augmented Generation for Textual Graph Question Answeringcited by 0
  12. 2026Topological Signal Processing: An Application-Oriented Tutorialcited by 0
  13. 2026Collapsed Effective Operators for Higher-order Structurescited by 0
  14. 2026Topology-Consistent Task Planning over Cellular Workflow Complexes for LLM-based Agentscited by 0
  15. 2026Batch Before You Lift: Scalable Topological Deep Learning on Large Graphscited by 0