A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.

翻译：ReLU神经网络导致输入空间的一个有限多面体分解及相应的有限对偶图。我们证明，虽然这个对偶图是输入空间的粗糙量化，但它足够鲁棒，可以结合持续同调从样本中检测输入空间中流形的同调信号。这一性质适用于多种为广泛目的而训练的网络，且这些训练目的与该拓扑应用无关。我们发现这一特性令人惊讶且有趣；我们也希望它能够具有实用价值。