Persistent homology (PH) is one of the main methods used in Topological Data Analysis. An active area of research in the field is the study of appropriate notions of PH representatives, which allow to interpret the meaning of the information provided by PH, making it an important problem in the application of PH, and in the study of its interpretability. Computing optimal PH representatives is a problem that is known to be NP-hard, and one is therefore interested in developing context-specific optimality notions that are computable in practice. Here we introduce time-optimal PH representatives for time-varying data, allowing one to extract representatives that are close in time in an appropriate sense. We illustrate our methods on quasi-periodic synthetic time series, as well as time series arising from climate models, and we show that our methods provide optimal PH representatives that are better suited for these types of problems than existing optimality notions, such as length-optimal PH representatives.

翻译：持续同调（PH）是拓扑数据分析中使用的主要方法之一。该领域的一个活跃研究方向是探究PH表示的适当概念，这些概念有助于解释PH所提供信息的含义，使其成为PH应用及其可解释性研究中的重要问题。计算最优PH表示是一个已知的NP难问题，因此研究者致力于开发可在实践中计算的情境特定最优性概念。本文针对时变数据引入时间最优PH表示，使得能够在适当意义上提取时间上接近的表示。我们在准周期合成时间序列以及气候模型产生的时间序列上展示了我们的方法，并证明相较于现有最优性概念（如长度最优PH表示），我们的方法能为这类问题提供更适用的最优PH表示。