Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and noise. Such properties are desirable in deep learning pipelines but they are typically obtained using non-TDA strategies. This is partly caused by the difficulty of combining TDA constructs (e.g. barcode and persistence diagrams) with current deep learning algorithms. Fortunately, we are now witnessing a growth of deep learning applications embracing topologically-guided components. In this survey, we review the nascent field of topological deep learning by first revisiting the core concepts of TDA. We then explore how the use of TDA techniques has evolved over time to support deep learning frameworks, and how they can be integrated into different aspects of deep learning. Furthermore, we touch on TDA usage for analyzing existing deep models; deep topological analytics. Finally, we discuss the challenges and future prospects of topological deep learning.

翻译：拓扑数据分析（TDA）能够揭示数据的形状特征。这些方法获得的摘要是对多维数据的原理性全局描述，同时具有稳定性质，例如对形变和噪声的鲁棒性。此类性质在深度学习管线中备受期待，但通常通过非TDA策略实现。这在一定程度上源于将TDA结构（如条形码和持续散点图）与当前深度学习算法结合的困难。幸运的是，我们正见证越来越多深度学习应用采用拓扑引导组件。本综述通过首先回顾TDA的核心概念，审视拓扑深度学习这一新兴领域。继而探究TDA技术的应用如何随时间演变以支持深度学习框架，以及它们如何融入深度学习的各个层面。此外，我们涉及TDA用于分析现有深度学习模型的应用，即深度拓扑分析。最后，我们讨论拓扑深度学习的挑战与未来前景。