Computational topology provides a tool, persistent homology, to extract quantitative descriptors from structured objects (images, graphs, point clouds, etc). These descriptors can then be involved in optimization problems, typically as a way to incorporate topological priors or to regularize machine learning models. This is usually achieved by minimizing adequate, topologically-informed losses based on these descriptors, which, in turn, naturally raises theoretical and practical questions about the possibility of optimizing such loss functions using gradient-based algorithms. This has been an active research field in the topological data analysis community over the last decade, and various techniques have been developed to enable optimization of persistence-based loss functions with gradient descent schemes. This survey presents the current state of this field, covering its theoretical foundations, the algorithmic aspects, and showcasing practical uses in several applications. It includes a detailed introduction to persistence theory and, as such, aims at being accessible to mathematicians and data scientists newcomers to the field. It is accompanied by an open-source library which implements the different approaches covered in this survey, providing a convenient playground for researchers to get familiar with the field.

翻译：计算拓扑学提供了一种工具——持续同调，用于从结构化对象（如图像、图、点云等）中提取定量描述符。这些描述符随后可被纳入优化问题中，通常作为引入拓扑先验或正则化机器学习模型的一种方式。这通常通过基于这些描述符最小化适当的、拓扑感知的损失函数来实现，而这自然引发了关于使用基于梯度的算法优化此类损失函数可能性的理论和实践问题。在过去十年中，这已成为拓扑数据分析领域的一个活跃研究方向，并且已开发出各种技术，使得通过梯度下降方案优化基于持续性的损失函数成为可能。本综述介绍了该领域的当前状态，涵盖其理论基础、算法层面，并展示了若干应用中的实际用途。它包含对持续性理论的详细介绍，并因此旨在可供该领域的新手——无论是数学家还是数据科学家——理解。本综述附带一个开源库，该库实现了综述中涵盖的不同方法，为研究人员熟悉该领域提供了一个便捷的平台。