准周期函数的滑动窗口持久性
Sliding Window Persistence of Quasiperiodic Functions
AI总结：
本文研究准周期函数滑动窗口点云的持久同调，提出参数优化方案并利用持久Künneth公式给出Rips持久同调理论下界，通过计算示例和音乐音频不协和检测验证了理论。
摘要：
如果一个函数的基本频率在有理数域上线性无关，则称其为准周期的。在适当参数下，此类函数的滑动窗口点云被证明在维度等于独立频率数量的环面中是稠密的。本文开发了理论与计算技术来研究此类集合的持久同调。具体而言，我们为准周期函数的滑动窗口提供了参数优化方案，并给出了其Rips持久同调的理论下界。后者利用了最近的持久Künneth公式。该理论通过计算示例及在音乐音频样本中不协和检测的应用得到了说明。
英文摘要：
A function is called quasiperiodic if its fundamental frequencies are linearly independent over the rationals. With appropriate parameters, the sliding window point clouds of such functions can be shown to be dense in tori with dimension equal to the number of independent frequencies. In this paper, we develop theoretical and computational techniques to study the persistent homology of such sets. Specifically, we provide parameter optimization schemes for sliding windows of quasiperiodic functions, and present theoretical lower bounds on their Rips persistent homology. The latter leverages a recent persistent Künneth formula. The theory is illustrated via computational examples and an application to dissonance detection in music audio samples.
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