  We propose a novel exact multi-parameter persistent homology method for
analyzing time-series data utilizing the Liouville torus. In the field of
topological data analysis (TDA), the conventional approach to analyzing
time-series data often involves sliding window embedding. From the perspective
of Takens' embedding theorem, we justify the analysis of the Liouville torus in
TDA and discuss the similarities and differences between the Liouville torus
and sliding window embedding approaches. We develop a multi-parameter
filtration method based on Fourier decomposition and provide an exact formula
of persistent homology with its one-parameter reduction of the multi-parameter
filtration. The conventional TDA of time-series data via sliding window is
known to be computationally expensive, but the proposed method yields the exact
barcode formula with the symmetry of the Liouville torus promptly, which
significantly reduces computational complexity while demonstrating comparable
or superior performance compared to the existing TDA methods. Furthermore, the
proposed method provides a way of obtaining various topological inferences by
exploring different filtration rays within the multi-parameter filtration
space, utilizing the nearly real-time computational capabilities of the
proposed method. The advantages of the proposed method significantly improve
the efficiency and flexibility of TDA when handling extensive time-series data
within machine learning workflows.
