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 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.

Figures (19)

• Figure 1: Left: Schematic model for a pendulum on a sliding block. Right: Illustration of anti-earthquake technology (see also PhysOrg2013Quake), whose dynamics are captured by the model on the left.
• Figure 2: Given a time series (left column), we reconstruct the underlying attractor via its sliding window embedding (middle column), and then compute its persistent homology (right column). Top row: Signal from the periodic motion of an ideal pendulum (left). The sliding window point cloud parametrizes a circle (middle), so its persistence diagram exhibits exactly one nontrivial 1-dimensional homology class (right). Bottom row: A quasiperiodic function with two incommensurate frequencies (left); its sliding window embedding (center) recovers a 2-torus. The corresponding persistence diagrams (right) confirm this topology by showing two independent 1-dimensional classes and one 2-dimensional class.
• Figure 3: Top row: The plot on the left is the phase space $(x,\dot{x})$ from the pendulum attached to a sliding block shown in Figure \ref{['fig:IlustPendulum']}. The solution $x(t)$ we used for the sliding window embedding is plotted in the middle, followed by the modulus of its Discrete Fourier Transform (right). Bottom row: Persistence diagrams of the sliding window embedding of $x(t)$(brown circles), of a truncated Fourier series of $x(t)$ (blue squares), and our proposed approximation $K_{3G}$ (orange crosses). The orange rectangles depict the theoretical approximation bound.
• Figure 4: Trajectories of the system from Example \ref{['ex:radial eq']}, plotted in the $xy$-plane (Left) and from Example \ref{['ex:van 2']} in the $x\dot{x}$ plane (Right). In each case, the attractor is a topological circle highlighted in red.
• Figure 5: Left column: the $x(t)$ coordinate of the solution to the radial system from Example \ref{['ex:radial eq']} (top row) and the van der Pol system from Example \ref{['ex:van 2']} (bottom row). Middle column: The sliding window embedding of $x(t)$ with appropriate parameters $d$ and $\tau$. Right column: The Rips persistence diagrams in dimensions 0,1, and 2 for the sliding window point cloud.

Theorems & Definitions (79)

• Remark 1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Definition 2.8
• Theorem 2.9