The time delay (or sliding-window) embedding is a technique from dynamical
systems to reconstruct attractors from time-series data. Recently, descriptors from
topological data analysis (TDA) — specifically, persistence diagrams — have been
used to measure the shape of said reconstructed attractors in applications including
periodicity and quasiperiodicity quantification. Despite their utility, the fast
computation of persistence diagrams of sliding-window embeddings is still
poorly understood. We present theoretical and computational schemes to
approximate the persistence diagrams of sliding-window embeddings from
quasiperiodic functions. We do so by combining the three-gap theorem from number
theory with the persistent Künneth formula from TDA, and derive fast
and provably correct persistent homology approximations. The input to our
procedure is the spectrum of the signal, and we provide numerical as well as
theoretical evidence of its utility to capture the shape of toroidal attractors.
Keywords
persistence diagrams, persistent homology, topological data
analysis, sliding-window embedding, quasiperiodic signals,
three-gap theorem, continued fractions, Rips filtration,
Künneth formula, time series analysis
