Quantum Persistent Homology for Time Series

TL;DR

A quantum Takens's delay embedding algorithm is established, which turns a time series into a point cloud by considering a pertinent embedding into a higher dimensional space.

Abstract

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.

Figures (6)

• Figure 1: Simplices of dimensions $k = 0, 1, 2, 3$.
• Figure 2: (a) The graph of $\sin(2\pi t)$ (blue line) along with a discrete time series (orange dots) given by $t = 0, 1/4, 1/2, 3/4, 1$. (b) The point cloud obtained by Takens's delay embedding using $\tau = 1$ and $d=2$.
• Figure 3: Vietoris Rips complexes of the embedded point cloud associated with the sinusodial signal of Fig. \ref{['fig:time-series-one']} at scales (a) $\epsilon_1 = 1.2$, the points are pairwise connected, which results in a hole in the middle; and (b) $\epsilon_2 = 2.2$, the points become totally connected and the hole disappears. The blue squares are the balls of diameter $\epsilon_{i}$ around each point.
• Figure 4: Persistence diagram for the time series in Fig \ref{['fig:time-series-one']}. The horizontal axis marks the scales at which the topological features are born, while the vertical axis marks the scales at which they disappear. The size of the dots represents the number of features that appear and disappear at the same scales. Finally, the orange dots along the vertical axis represent the connected components, while the blue dots closer to the diagonal line are the one dimensional holes.
• Figure 5: (a) A segment of an EEG signal measured while the subject listened to music eeg-data. (b) The point cloud obtained by the Takens's delay embedding using $\tau = 8$ and $d=2$.