Topology applied to real world data using persistent homology has started to
find applications within machine learning, including deep learning. We present
a differentiable topology layer that computes persistent homology based on
level set filtrations and edge-based filtrations. We present three novel
applications: the topological layer can (i) regularize data reconstruction or
the weights of machine learning models, (ii) construct a loss on the output of
a deep generative network to incorporate topological priors, and (iii) perform
topological adversarial attacks on deep networks trained with persistence
features. The code (www.github.com/bruel-gabrielsson/TopologyLayer) is publicly
available and we hope its availability will facilitate the use of persistent
homology in deep learning and other gradient based applications.

使用可持续同调应用于拓扑学于机器学习，包括深度学习。该研究提出了一种可微拓扑层，该层基于水平集过滤和边缘过滤计算持续同调。该层在数据重建或机器学习模型权重的规范化，构建构成深度生成网络输出的损失函数以包括拓扑先验，以及对使用持久性特征训练的深度网络进行拓扑对抗攻击等三种应用方面具有创新性。代码公开可用，有望促进深度学习和其他梯度基础应用中的持久同调的使用。