We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively, from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

我们定义了从和到持久性条形码空间的映射的可区分性概念。受差异空间理论的启发，所提出的框架使用提升到有序条形码空间，从中可以计算导数。可微性的两个派生概念（分别是从和到条形码空间）自然地结合在一起以产生一个链式法则，该链式法则允许使用梯度下降来分解通过条形码空间的目标函数。我们通过展示如何使用它来分析拓扑数据分析中出现的各种参数化过滤系列的平滑度来说明该框架的多功能性。