\v{C}ech-Delaunay gradient flow and homology inference for self-maps

TL;DR

This paper introduces a novel method combining ccch-Delaunay gradient flow and persistent homology to analyze sampled dynamical systems, enabling efficient homology inference and eigenspace recovery.

Contribution

It presents a new sampling theorem for recovering homology eigenspaces and a combinatorial gradient flow approach that efficiently transforms ccch complexes into Delaunay complexes for homology analysis.

Findings

Established a sampling theorem for eigenspace recovery.

Developed an efficient chain map from ccch to Delaunay complexes.

Demonstrated applications of the chain map beyond dynamical systems.

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. We establish a sampling theorem to recover the eigenspace of the endomorphism on homology induced by the self-map. Using a combinatorial gradient flow arising from the discrete Morse theory for \v{C}ech and Delaunay complexes, we construct a chain map to transform the problem from the natural but expensive \v{C}ech complexes to the computationally efficient Delaunay triangulations. The fast chain map algorithm has applications beyond dynamical systems.