The time delay (or Sliding Window) embedding is a technique from dynamical systems to reconstruct attractors from time series data. Recently, descriptors from Topological Data Analysis (TDA) -- specifically, persistence diagrams -- have been used to measure the shape of said reconstructed attractors in applications including periodicity and quasiperiodicity quantification. Despite their utility, the fast computation of persistence diagrams of sliding window embeddings is still poorly understood. In this work, we present theoretical and computational schemes to approximate the persistence diagrams of sliding window embeddings from quasiperiodic functions. We do so by combining the Three Gap Theorem from number theory with the Persistent Künneth formula from TDA, and derive fast and provably correct persistent homology approximations. The input to our procedure is the spectrum of the signal, and we provide numerical as well as theoretical evidence of its utility to capture the shape of toroidal attractors.

翻译：时间延迟（或滑动窗口）嵌入是一种从时间序列数据重构吸引子的动力学系统技术。近年来，拓扑数据分析（TDA）中的描述符——特别是持续性图——已被用于测量此类重构吸引子的形状，其应用包括周期性和准周期性量化。尽管这些方法具有实用性，但滑动窗口嵌入持续性图的快速计算仍鲜有研究。在本工作中，我们提出了从准周期函数近似计算滑动窗口嵌入持续性图的理论与计算方案。为此，我们将数论中的三间隙定理与TDA中的持续性Künneth公式相结合，推导出快速且可证明正确的持续性同调近似方法。我们算法的输入是信号的频谱，并通过数值与理论证据证明了该方法在捕捉环面吸引子形状方面的有效性。