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Exact multi-parameter persistent homology of time-series data: one-dimensional reduction of multi-parameter persistence theory

Keunsu KimJae-Hun Jung
Nov 2022
摘要
In various applications of data classification and clustering problems,multi-parameter analysis is effective and crucial because data are usuallydefined in multi-parametric space. Multi-parameter persistent homology, anextension of persistent homology of one-parameter data analysis, has beendeveloped for topological data analysis (TDA). Although it is conceptuallyattractive, multi-parameter persistent homology still has challenges in theoryand practical applications. In this study, we consider time-series data and itsclassification and clustering problems using multi-parameter persistenthomology. We develop a multi-parameter filtration method based on Fourierdecomposition and provide an exact formula and its interpretation ofone-dimensional reduction of multi-parameter persistent homology. The exactformula implies that the one-dimensional reduction of multi-parameterpersistent homology of the given time-series data is equivalent to choosingdiagonal ray (standard ray) in the multi-parameter filtration space. For this,we first consider the continuousization of time-series data based on Fourierdecomposition towards the construction of the exact persistent barcode formulafor the Vietoris-Rips complex of the point cloud generated by sliding windowembedding. The proposed method is highly efficient even if the sliding windowembedding dimension and the length of time-series data are large because themethod precomputes the exact barcode and the computational complexity is as lowas the fast Fourier transformation of $O(N \log N)$. Further the proposedmethod provides a way of finding different topological inferences by tryingdifferent rays in the filtration space in no time.
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