用于时间序列的量子持续同调
Quantum Persistent Homology for Time Series
AI总结：
针对现有量子持续同调仅支持点云数据的限制，本文提出量子 Takens 延迟嵌入算法，将时间序列转化为高维点云，从而可利用量子持续同调提取其拓扑特征。
摘要：
持续同调是一种强大的数据分析数学工具，它通过追踪不同尺度变化下的拓扑特征来概括数据的形状。持续同调的经典算法往往受到运行时间和内存需求随数据点数量呈指数增长的限制。为克服这一问题，已有两种基于不同方法的持续同调量子算法被开发出来。然而，这两种量子算法都以点云形式的数据集为处理对象；考虑到许多数据集以时间序列形式出现，这种假设具有局限性。在本文中，我们通过建立一种量子 Takens 延迟嵌入算法来缓解这一问题，该算法通过将时间序列适当地嵌入到更高维空间中，将其转化为点云。有了这种从时间序列到点云的量子变换后，便可以使用量子持续同调算法，从与原始时间序列相关联的点云中提取拓扑特征。
英文摘要：
Persistent homology, a powerful mathematical tool for data analysis, summarizes the shape of data through tracking topological features across changes in different scales. Classical algorithms for persistent homology are often constrained by running times and memory requirements that grow exponentially on the number of data points. To surpass this problem, two quantum algorithms of persistent homology have been developed based on two different approaches. However, both of these quantum algorithms consider a data set in the form of a point cloud, which can be restrictive considering that many data sets come in the form of time series. In this paper, we alleviate this issue by establishing a quantum Takens's delay embedding algorithm, which turns a time series into a point cloud by considering a pertinent embedding into a higher dimensional space. Having this quantum transformation of time series to point clouds, then one may use a quantum persistent homology algorithm to extract the topological features from the point cloud associated with the original times series.
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