We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture: namely, permutation equivariance, orientation equivariance, and simplicial awareness. The first two properties respectively account for the fact that the node indexing and the simplex orientations in a simplicial complex are arbitrary. The last property encodes the desirable feature that the output of the neural network depends on the entire simplicial complex and not on a subset of its dimensions. Based on these properties, we propose a simple convolutional architecture, rooted in tools from algebraic topology, for the problem of trajectory prediction, and show that it obeys all three of these properties when an odd, nonlinear activation function is used. We then demonstrate the effectiveness of this architecture in extrapolating trajectories on synthetic and real datasets, with particular emphasis on the gains in generalizability to unseen trajectories.

中文翻译：

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原理简单的神经网络用于轨迹预测

我们考虑针对简单复合物的数据构建神经网络体系结构。在研究简单复数的链复数上的图时，我们定义了简单神经网络体系结构的三个理想属性：即置换等方差，方向等方差和简单知觉。前两个属性分别说明了以下事实：单纯形复数中的节点索引和单纯形方向是任意的。最后一个属性编码了一个理想的特征，即神经网络的输出取决于整个简单复数而不是其维数的子集。基于这些特性，我们提出了一种简单的卷积架构，该算法以代数拓扑为工具，针对轨迹预测问题，并表明当使用奇数非线性激活函数时，它遵循所有这三个属性。然后，我们证明了该体系结构在合成和实际数据集上外推轨迹时的有效性，并特别强调了对看不见的轨迹的通用性方面的收益。