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.

翻译：我们提出了一种单纯复形卷积神经网络（SCCNN），用于在单纯复形上学习数据表示。该方法基于多跳单纯邻接关系，通过共用面和余面独立执行卷积操作，捕获单纯间耦合，从而推广了现有最优方法。通过对单纯域和数据空间的对称性研究，证明了该网络具有置换等变和定向等变性，从而融入了此类归纳偏置。基于霍奇理论，我们进行了频谱分析以理解SCCNN如何调控不同频率的数据，表明通过面和余面进行的卷积在两个正交数据空间中运行。最后，我们研究了SCCNN对域变形的稳定性，并考察了各种因素的影响。实验结果表明，高阶卷积和单纯间耦合在单纯预测和轨迹预测中具有优势。