Simplicial map neural networks (SMNNs) are topology-based neural networks with interesting properties such as universal approximation ability and robustness to adversarial examples under appropriate conditions. However, SMNNs present some bottlenecks for their possible application in high-dimensional datasets. First, SMNNs have precomputed fixed weight and no SMNN training process has been defined so far, so they lack generalization ability. Second, SMNNs require the construction of a convex polytope surrounding the input dataset. In this paper, we overcome these issues by proposing an SMNN training procedure based on a support subset of the given dataset and replacing the construction of the convex polytope by a method based on projections to a hypersphere. In addition, the explainability capacity of SMNNs and an effective implementation are also newly introduced in this paper.

翻译：单纯复形映射神经网络（SMNNs）是一种基于拓扑结构的神经网络，具有在适当条件下实现通用逼近能力和对对抗样本鲁棒性的优良特性。然而，SMNNs在高维数据集的应用中面临两大瓶颈：首先，SMNNs使用预计算固定权重，至今未定义任何训练过程，因此缺乏泛化能力；其次，SMNNs需要构建包围输入数据集的凸多面体。本文通过提出基于给定数据集支持子集的SMNN训练流程，并将凸多面体构建方法替换为基于超球投影的方案，成功解决了上述问题。此外，本文还首次引入SMNN的可解释性能力及其实效性实现方案。