Extended persistence is a technique from topological data analysis to obtain global multiscale topological information from a graph. This includes information about connected components and cycles that are captured by the so-called persistence barcodes. We introduce extended persistence into a supervised learning framework for graph classification. Global topological information, in the form of a barcode with four different types of bars and their explicit cycle representatives, is combined into the model by the readout function which is computed by extended persistence. The entire model is end-to-end differentiable. We use a link-cut tree data structure and parallelism to lower the complexity of computing extended persistence, obtaining a speedup of more than 60x over the state-of-the-art for extended persistence computation. This makes extended persistence feasible for machine learning. We show that, under certain conditions, extended persistence surpasses both the WL[1] graph isomorphism test and 0-dimensional barcodes in terms of expressivity because it adds more global (topological) information. In particular, arbitrarily long cycles can be represented, which is difficult for finite receptive field message passing graph neural networks. Furthermore, we show the effectiveness of our method on real world datasets compared to many existing recent graph representation learning methods.

翻译：扩展持续性是拓扑数据分析中的一种技术，用于从图中获取全局多尺度拓扑信息，包括由所谓持续条码捕获的连通分量和环的信息。我们将扩展持续性引入图分类的监督学习框架中。全局拓扑信息（以包含四种不同类型条及其显式环代表元的条码形式）通过由扩展持续性计算的读出函数整合到模型中。整个模型是端到端可微的。我们采用链接-割树数据结构与并行计算来降低扩展持续性计算复杂度，相比现有最先进的扩展持续性计算方法实现了超过60倍的加速，从而使其在机器学习中具有可行性。我们证明，在特定条件下，扩展持续性在表达能力上超越了WL[1]图同构测试和零维条码，因为它融入了更多全局（拓扑）信息。特别地，该方法能够表示任意长度的环——这对有限感受野的消息传递图神经网络而言是困难的。此外，在真实数据集上，与多种现有图表示学习方法相比，我们的方法展现出有效性。