Convolutional Learning on Simplicial Complexes

2023/01/26 by Maosheng Yang, Yang, Maosheng, Elvin Isufi +1 · 4 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · #Bioinformatics and Genomic Networks #FOS: Computer and information sciences #Machine Learning (cs.LG) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2301.11163

openalex publication_date 2023/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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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• Topological Adaptive Least Mean Squares Algorithms over Simplicial Complexes
• Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding
• HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

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