[Paper Review] Persistence in sampled dynamical systems faster.

Ulrich Bauer, Herbert Edelsbrunner+2 more|arXiv (Cornell University)|Sep 12, 2017

Topological and Geometric Data Analysis3 citations

TL;DR

This paper introduces a fast algorithm for analyzing sampled dynamical systems by using persistent homology on Delaunay complexes to detect recurrent behavior and recover homological eigenspaces. It leverages discrete Morse theory to implicitly represent discrete gradient fields, enabling efficient chain map computations with reduced computational overhead.

ABSTRACT

We call a continuous self-map that reveals itself through a discrete set of point-value pairs a sampled dynamical system. Capturing the available information with chain maps on Delaunay complexes, we use persistent homology to quantify the evidence of recurrent behavior, and to recover the eigenspaces of the endomorphism on homology induced by the self-map. The chain maps are constructed using discrete Morse theory for Cech and Delaunay complexes, representing the requisite discrete gradient field implicitly in order to get fast algorithms.

Motivation & Objective

• To develop an efficient method for analyzing continuous self-maps observed only through discrete point-value pairs.
• To quantify recurrent behavior in sampled dynamical systems using topological data analysis.
• To recover the eigenspaces of the endomorphism on homology induced by the self-map.
• To accelerate persistent homology computations via implicit discrete gradient fields on Cech and Delaunay complexes.

Proposed method

• Constructing chain maps on Delaunay complexes from sampled point-value data to represent the dynamics.
• Applying persistent homology to detect and quantify recurrent behavior in the sampled system.
• Using discrete Morse theory to implicitly encode the discrete gradient field, avoiding explicit construction of gradient vector fields.
• Representing the dynamics via chain maps that preserve topological structure across the Delaunay complex.
• Leveraging the implicit gradient field to reduce computational cost in persistent homology computations.
• Mapping the induced endomorphism on homology to recover eigenspaces associated with the self-map.

Experimental results

Research questions

• RQ1 How can recurrent behavior in sampled dynamical systems be quantified using topological invariants?
• RQ2 What is the role of Delaunay complexes in preserving topological structure from discrete samples?
• RQ3 How can discrete Morse theory be used to accelerate persistent homology computations in dynamical systems?
• RQ4 Can the eigenspaces of the homological endomorphism be recovered from sampled data using persistent homology?
• RQ5 What is the computational advantage of using implicit gradient fields in chain map construction?

Key findings

• The method successfully detects and quantifies recurrent behavior in sampled dynamical systems using persistent homology on Delaunay complexes.
• The use of discrete Morse theory enables efficient computation by implicitly representing the discrete gradient field, reducing algorithmic complexity.
• The chain maps constructed on Delaunay complexes preserve the topological structure necessary to recover homological eigenspaces.
• The approach achieves faster computation compared to explicit gradient field constructions, without sacrificing topological fidelity.
• Persistent homology applied to sampled systems reveals evidence of recurrence even when the underlying dynamics are not directly observable.
• The framework supports the recovery of the endomorphism's eigenspaces on homology, providing insight into the system's long-term behavior.