Sliding window persistence of quasiperiodic functions
Sliding window persistence of quasiperiodic functions
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DOI：
10.1007/s41468-023-00136-7
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
H. Gakhar;Jose A. Perea
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文献类型：
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作者：
H. Gakhar;Jose A. Perea
如果一个函数的基频与有理数线性无关，则该函数被称为准周期函数。通过适当的参数，此类函数的滑动窗口点云可以显示为密集的环面，其尺寸等于独立频率的数量。在本文中，我们开发了理论和计算技术来研究此类集合的持久同源性。具体来说，我们提供了准周期函数滑动窗口的参数优化方案，并提出了其 Rips 持久同源性的理论下界。后者利用了最近持久的 Künneth 公式。通过计算示例和音乐音频样本中不和谐检测的应用来说明该理论。
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.