Estimation of Persistence Diagrams via the Three Gap Theorem

Abstract

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. In this work, 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 K\"unneth 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.