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Convolutional Learning on Simplicial Complexes

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arxiv 2301.11163 v1 pith:AYTRSIUI submitted 2023-01-26 cs.LG

Maosheng Yang , Elvin Isufi This is my paper · ORCID
classification cs.LG
keywords simplicialdataconvolutionscofacescomplexesconvolutionalcouplingsdomain
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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.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Geometry-Induced Hodge Stars on Rips and Dowker--Rips Complexes

    math.AT 2026-07 conditional novelty 6.0 of 10

    A weighted Hodge Laplacian on Rips-type complexes with volume- or witness-based diagonal weights preserves Betti numbers and yields geometry-sensitive spectra.

  2. Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes

    eess.SP 2025-05 conditional novelty 6.0 of 10

    Topo-LMS brings classical least-mean-squares adaptive filtering to edge signals over simplicial complexes, with stability conditions, closed-form steady-state error, and optimal sampling strategies.

  3. HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

    cs.LG 2025-05 conditional novelty 6.0 of 10

    HOPSE encodes higher-order topological data by applying graph positional and structural encoders to Hasse graph decompositions, matching or exceeding message-passing models on benchmarks with up to 7x faster training.

  4. Heat Kernel Goes Topological

    cs.LG 2025-07 reject novelty 5.0 of 10

    TopoHKS defines a weighted combinatorial-complex Laplacian and heat kernel descriptor, claiming maximal expressive power; the supporting uniqueness theorem is incorrect.

  5. CellCLAT: Preserving Topology and Trimming Redundancy in Self-Supervised Cellular Contrastive Learning

    cs.LG 2025-05 conditional novelty 5.0 of 10

    CellCLAT applies parameter-perturbation contrastive learning to cellular complexes and adaptively trims 2-cells to improve downstream graph classification.

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